UC-NRLF 


•- 


ELEMENTS 

OF  THE 

PRECISION  OF  MEASUREMENTS 

AND 

GRAPHICAL  METHODS 


BY 

H.  M.  GOODWIN,  PH.D. 

PROFESSOR  OF  PHYSICS 
MASSACHUSETTS  INSTITUTE   OF  TECHNOLOGY 


McGRAW-HILL   BOOK    COMPANY 

239  WEST  39-TH  STREET,  NEW  YORK 

6  Bouverie  Street,  London,  E.C. 

1913 


Engineering 
Library 


COPYRIGHT,  1908 

REVISED,  EXTENDED,  AND  COPYRIGHTED,  1913 
BY  H.  M.  GOODWIN 

All  rights  reserved 


PRESS  OF  GEO.  H.  ELLIS  CO.,  BOSTOM 


PREFACE. 


In  its  present  form  the  "Elements  of  the  Precision  of  Measure- 
ments and  Graphical  Methods"  represents  the  ground  covered  in  a 
brief  course  which  has  been  given  for  a  number  of  years  at  the  Massa- 
chusetts Institute  of  Technology  to  all  students  in  connection  with 
their  work  in  the  Physical  Laboratory.  The  author  has  been  induced 
to  amplify  the  printed  "Notes"  on  this  subject  and  give  them  a  wider 
circulation  in  response  to  repeated  requests  to  use  them  elsewhere. 
Although  prepared  primarily  to  meet  the  needs  of  his  own  classes,  it 
is  hoped,  in  the  present  form,  they  may  prove  useful  in  other  tech- 
nical schools  and  colleges  where  quantitative  work  forms  a  part  of 
the  curriculum,  and  also  to  engineers  whose  work  involves  experi- 
mental testing.  In  many  laboratories  far  too  little  weight  is  attached 
to  the  discussion  of  the  magnitude  and  effect  of  sources  of  error 
on  a  result.  This  has  been  forced  upon  the  writer's  attention  as  the 
result  of  personal  interviews  with  hundreds  of  graduate  students 
entering  the  Institute,  who  apply  for  excuse  from  laboratory  work. 
It  is  the  exceptional  student  who  has  any  conception  how  to  figure 
out  the  precision  of  a  final  computed  result  from  the  precision 
of  his  individual  measurements,  and  this  is  true  even  though  his 
laboratory  note-book  shows  his  work  to  have  been  carefully  and  credi- 
tably performed.  It  is  the  author's  firm  conviction  that  one  of  the 
most  valuable  and  enduring  benefits  of  physical  laboratory  training 
to  a  student  of  Science  or  Engineering  is  the  acquisition  of  the  proper 
view-point  with  which  to  approach  an  investigation,  be  it  either 
purely  scientific  or  technical;  that  is,  the  ability  to  recognize  the 
essentials  of  a  problem  at  the  outset,  so  as  to  economize  both  time 
and  labor  in  its  solution.  Although  the  exercise  of  judgment,  based 
upon  the  personal  experience  of  the  investigator,  is  essential  to  the 
"best  solution"  of  any  experimental  problem,  still  it  is  desirable  to 
direct  the  student's  attention  to  precision  methods  at  an  early  stage 
of  his  laboratory  work.  Experience  has  shown  that  this  may  be  satis- 
factorily done  as  soon  as  he  has  had  a  little  practice  in  exact  measure- 
ments and  can  handle  the  elements  of  Differential  Calculus.  At  the 
Institute  the  course  is  given  at  the  middle  of  the  sophomore  year, 
after  the  student  has  performed  some  six  or  eight  experiments  on 
fundamental  measurements  in  Mechanics.  Continued  application  of 
the  principles  is  then  made  in  subsequent  laboratory  work  throughout 
the  junior  and  senior  years,  and  a  precision  discussion  is  regarded  as 


267862 


PREFACE 

an  important  feature  of  the  final  thesis.  It  has  been  the  writer's 
experience  that  students  have  little  trouble  in  understanding  the  gen- 
eral principles  involved,  but  meet  with  considerable  difficulty  in 
applying  these  principles  to  concrete  problems.  For  this  reason  the 
subject  is  most  satisfactorily  taught  to  small  sections  by  recitations 
based  on  the  text  and  the  solution  otnumerous  problems  selected  from 
the  book  and  from  the  current  laboratory  work.  A  close  correlation 
of  class-room  and  laboratory  work  is  indeed  highly  desirable,  and  in 
the  Rogers  Laboratory  of  Physics  it  is  the  practice  to  require  with 
each  laboratory  report  a  precision  discussion  of  the  data  or  a  solution 
of  some  precision  problem  related  to  the  experiment.  The  laboratory 
manuals  have  been  written  with  this  in  view. 

The  method  of  treatment  has  been  kept  as  brief  as  possible.  A 
full  discussion  of  the  subject,  with  proofs  based  on  the  Theory  of 
Probability  and  the  Method  of  Least  Squares,  would  so  enlarge  the 
work  as  to  defeat  its  end.  Proofs  of  the  few  theorems  and  formula 
which  the  student  is  asked  to  assume  may  be  found  in  any  good  treatise 
on  Least  Squares.  An  excellent  treatment  is  that  given  in  Bartlett's 
"Method  of  Least  Squares."  A  more  exhaustive  treatment  of  Pre- 
cision Methods-  may  be  found  in  Holman's  "Precision  of  Measure- 
ments." 

A  chapter  on  the  solution  of  illustrative  problems  has  been  added 
to  assist  students  who  find  it  necessary  to  work  up  the  subject  by 
themselves.  The  collection  of  problems  has  been  compiled  from 
recent  examination  papers.  The  chapter  on  Graphical  Methods  con- 
tains specific  directions  for  constructing  graphs,  and  general  directions 
for  obtaining  therefrom  the  functional  relationship  between  two 
variables.  For  engineering  students,  as  well  as  physicists,  the  method 
of  logarithmic  plotting  will  be  found  of  wide  application.  In  the 
Appendix  several  tables,  of  assistance  in  precision  computations, 
have  been  added. 

In  conclusion  the  author  desires  to  express  his  indebtedness  for 
many  suggestions  to  his  colleagues  who  have  so  ably  assisted  him  in 
the  instruction  of  this  subject  in  recent  years,  and  in  particular  to 
Professor  William  J.  Drisko,  whose  experience  in  teaching  this  and 
related  subjects  has  been  most  helpful. 

H.  M.  GOODWIN. 


CONTENTS. 

PART   I. 
PRECISION  OF  MEASUREMENTS. 

Classification  of  Measurements 7 

Precision  Discussion  of  Direct  Measurements 7 

Determinate  Errors 8 

Indeterminate  Errors 11 

Law  of  Error 13 

Method  of  Least  Squares 14 

Deviation  and  Precision  Measures 15 

Average  Deviation  of  a  Single  Observation 16 

Deviation  of  the  Mean 16 

Fractional  and  Percentage  Deviation 17 

Deviation  vs.  Precision  Measures 17 

Probable  Error v    .  18 

Mean  Error 19 

Weights  and  Weighted  Mean 20 

Criterion  for  Rejecting  Observations 20 

Computation  Rules  and  Significant  Figures 21 

Precision  Discussion  of  Indirect  Measurements 25 

Separate  Effects 27 

Resultant  Effects 29 

Criterion  for  Negligibility  of  an  Error 30 

Equal  Effects 32 

Fractional  Method  of  Solution    .  33 


PART  II. 

GRAPHICAL  METHODS. 

Nature  of  Problems 41 

Procedure  in  Plotting 41 

Determination  of  Constants  of  a  Straight  Line 46 

Rectification  of  Curved  Lines 47 

Trigonometric  Functions 48 

Reciprocal  Functions 50 


6  CONTENTS 

PAGE 

The  Logarithmic  Method.    Exponential  Functions;  y  =  mxn  .    ...  52 
Special  Cases: 

y  =  m(x  +  p)"> 58 

y  =  m  IQn*;  y  =  menx 59 

Precision  of  Plotting 59 

Residual  Plots , 60 

Interpolation  Formulae ' 63 

Graphical  Solution 64 

Least  Square  Solution 64 


PART  III. 
PROBLEMS. 

Solution  of  Illustrative  Problems 69 

Problems  81 


APPENDIX. 

Table     I.     Mathematical  Constants 99 

Table    II.    Approximation  Formulae 100 

Table  III.     Squares,  Cubes,  and  Reciprocals 101 

Table  IV.    Four  Place  Logarithms 102 

Table    V.    Sines,  Cosines,  and  Tangents 104 


PRECISION   OF   MEASUREMENTS. 


Classification  of  Physical  Measurements. — All  physical 
measurements  may  be  classed  as  direct  or  indirect  ac- 
cording as  the  measurement  gives  the  desired  result  di- 
rectly, or  as  the  result  is  obtained  by  combining  the  re- 
sults of  several  measurements  by  means  of  some  formula. 
Examples  of  the  first  class  are  the  measurement  of  a  length 
by  means  of  a  scale,  the  mass  of  a  body  by  means  of  an  equal 
arm  balance,  and  the  electrical  resistance  of  a  wire  by  the 
direct  method  of  substitution.  Examples  of  indirect  meas- 
urements are  the  determination  of  g,  the  acceleration  due  to 
gravity,  by  means  of  a  pendulum,  involving  the  measure- 
ment of  the  length  and  time  of  vibration  of  the  pendulum, 
the  determination  of  the  index  of  refraction  of  a  substance 
from  measurements  of  the  angle  and  the  minimum  deviation 
of  a  prism  by  means  of  a  spectrometer,  and  the  determina- 
tion of  the  specific  heat  of  a  substance  by  the  method  of 
mixtures  hi  which  the  results  of  the  measurement  of  a  num- 
ber of  temperatures  and  weights  are  combined.  The  great 
majority  of  problems  arising  hi  practice  come  under  the 
second  class. 

Reliability  of  a  Result.— In  order  that  the  result  of  any 
measurement,  whether  direct  or  indirect,  may  be  of  any 
scientific  or  technical  value,  it  is  necessary  to  have  some 
numerical  estimate  or  measure  of  its  reliability.  The  im- 
portance of  such  a  measure  cannot  be  overestimated.  The 
result  of  a  test,  of  a  study  of  an  instrument  or  method,  or 
of  the  determination  of  a  constant,  may  be  rendered  almost 
worthless,  unless  the  investigator  is  able  to  state  the  degree 
of  reliance  which  can  be  placed  upon  it.  This  phase  of 
an  investigation  should  be  kept  constantly  hi  mind  in  all 
laboratory  work.  The  student's  ability  to  intelligently  dis- 


8  PRECISION    OF    MEASUREMENTS 

cuss  the  reliability  of  his  data  is  regarded  as  of  no  less  im- 
portance than  his  ability  to  perform  accurate  work. 

By  the  precision  or  precision  measure  of  a  result,  denoted 
for  brevity  by  p.m.,  will  be  always  understood  the  best  nu- 
merical measure  of  its  reliability  which  can  be  obtained  after 
all  known  sources  of  error  have  t>een  eliminated  or  corrected 
for.  How  this  may  be  computed  will  be  explained  below. 
By  the  accuracy  of  a  result  should,  strictly  speaking,  be  un- 
derstood the  degree  of  concordance  between  it  and  the  true 
value  of  the  quantity  measured.  Since,  however,  the  latter 
is  usually  unknown,  it  is  seldom  that  we  can  obtain  a  numeri- 
cal measure  of  the  absolute  accuracy  of  a  measurement.  We 
must  in  most  cases  be  content  with  an  estimated  or  computed 
precision  measure.  The  terms  "accuracy"  and  "precision" 
are  often  carelessly  used  indiscriminately. 

The  precision  measure  of  a  direct  measurement  is  of  no 
less  importance  than  of  an  indirect  measurement.  As  the 
precision  of  the  latter  depends  primarily  on  the  precision  of 
the  separate  components  from  which  it  is  computed,  the 
method  of  determining  a  numerical  estimate  of  the  reliability 
of  a  series  of  direct  observations  wrill  first  be  considered. 

Classification  of  Errors. — When  any  quantity  is  measured 
to  the  full  precision  of  which  the  instrument  or  method  em- 
ployed is  capable,  it  will  in  general  be  found  that  the  results 
of  repeated  measurements  do  not  exactly  agree.  This  is 
true  not  only  of  results  obtained  by  different  observers  using 
different  instruments  and  methods,  but  also  when  the  meas- 
urements are  made  by  the  same  observer  under  similar  con- 
ditions. The  cause  of  these  discrepancies  lies  in  various 
sources  of  error  to  which  all  experimental  data  are  subject. 
These  may  be  grouped  conveniently  in  two  classes, — de- 
terminate and  indeterminate  errors. 

Determinate  Errors. — Determinate  errors  are,  as  their 
name  indicates,  of  such  a  nature  that  their  value  can  be  de- 
termined and  their  effect  on  the  result  thereby  eliminated. 
They  may  be  classified  as  follows: — 

a.    Instrumental  Errors. — These  may  arise  from  poor  con- 


DETERMINATE    ERRORS  9 

struct  ion  or  faulty  adjustment  of  an  instrument,  as,  for  ex- 
ample, a  defect  in  a  micrometer  screw,  faulty  graduations 
of  scales  and  circles,  eccentricity  of  circles,  unequal  balance 
arms,  etc. 

b.  Personal  Errors. — These  may  arise  from  characteristic 
peculiarities  of  individual  observers,  as,  for  example,   the 
tendency  to  always  record  the  occurrence  of  an  event  too 
soon   or   too   late.     This   frequently  happens   in  recording 
transit  observations  in  which  the  "personal  equation"  of  the 
observer  becomes  an  important  factor. 

c.  Errors  of  Method  or   Theoretical  Errors. — These  may 
arise  from  using  an  instrument  under  conditions  for  which 
its  graduations  are  not  standard. 

The  following  illustrations  will  make  clearer  the  nature  of 
the  above  sources  of  error.  Suppose  that  the  arms  of  a 
chemical  balance  are  slightly  unequal  in  length.  All  weigh- 
ings made  with  such  a  balance  will  be  in  error  due  to  this 
cause  (if  the  balance  be  used  in  the  ordinary  way),  by  an 
amount  depending  on  the  inequality  in  the  length  of  the 
arms.  Repeated  weighings  of  the  same  substance  on  the 
same  balance  by  the  same  method  will,  however,  give  no 
indication  of  the  presence  of  this  source  of  error.  The  re- 
peated independent  weighings  may  indeed  check  among 
themselves  to  the  full  sensitiveness  of  the  balance,  and  yet 
the  result  may  be  in  error,  due  to  the  constant  instrumental 
error,  by  a  very  large  amount.  The  presence  of  such  an  error 
would  only  be  detected  by  comparing  the  results  of  the  weight 
of  the  same  body  obtained  on  different  balances  or  by  differ- 
ent methods  of  weighing,  for  the  probability  of  the  same  in- 
strumental error  occurring  to  the  same  extent  in  different 
instruments' is  very  small. 

Again,  suppose  a  length  is  measured  by  means  of  a  grad- 
uated scale  at  20°  C.,  while  the  scale  is  standard  at  some 
other  temperature,  say  0°  C.  Repeated  measurements  with 
such  a  scale  by  the  same  method  and  under  the  same  condi- 
tions would  probably  show  a  very  close  agreement  among 
themselves,  and  give  no  clew  to  the  presence  of  any  constant 


10  PRECISION   OF   MEASUREMENTS 

error.  The  result  would,  however,  be  too  small,  since  the 
value  of  the  units  of  the  scale  would  all  be  too  large,  due  to 
the  expansion  of  the  scale  from  0°  to  20°.  The  error  thus 
introduced  by  using  the  scale  under  conditions  other  than 
those  for  which  it  is  standard  is,  however,  determinate  in  its 
nature,  since  a  knowledge  of  the  coefficient  of  expansion  of 
the  scale  and  of  the  temperature  at  which  it  is  standard, 
and  also  at  which  it  is  used,  furnishes  all  necessary  data, 
for  reducing  the  observed  result  to  the  value  it  would 
have  had,  had  the  scale  been  standard  at  the  time  of  the 
measurement.  The  concordance  of  a  series  of  observations 
taken  under  similar  conditions  is,  therefore,  no  criterion  of 
the  absence  of  constant  errors  even  of  very  large  amount. 

To  detect  and  eliminate  such  errors,  it  is  necessary  to  com- 
pare the  results  of  measurements  of  the  quantity  by  different 
methods,  different  apparatus,  and,  if  possible,  different  ob- 
servers, and  to  average  such  independent  results  by  a  special 
method  described  later;  for  the  probability  of  the  same  source 
of  error  being  present  under  such  variable  conditions  is  very 
small.  An  interesting  illustration  of  the  presence  of  a  con- 
stant error  escaping  detection  is  to  be  found  in  the  original 
determination  of  the  ohm  by  the  British  Association  Com- 
mittee. The  excellent  agreement  of  the  observations  among 
themselves  lead  to  the  conclusion  that  the  result  possessed  a 
high  degree  of  reliability.  Later  determinations  by  inde- 
pendent methods  and  observers  gave  values  which  differed 
from  the  B.  A.  value  by  over  one  per  cent.,  an  amount  far 
hi  excess  of  the  precision  with  which  the  B.  A.  determination 
had  been  carried  out.  Attention  was  thus  called  to  the 
probable  presence  of  some  constant  error  which  further  in- 
vestigation verified. 

Residual  Errors. — After  a  result  has  been  corrected  as  well 
as  may  be  for  all  known  sources  of  determinate  errors,  there 
may  still  remain  in  it  small  errors,  the  value  of  which  cannot 
be  determined,  and  which,  therefore,  fall  into  the  second 
general  class  of  errors, — indeterminate  errors.  Thus,  if  the 
instrumental  error  arising  from  inequality  in  the  length  of 


INDETERMINATE    ERRORS  11 

balance  arms  be  corrected  by  a  determination  of  the  ratio 
of  the  arms,  this  ratio  will  be  known  to  only  a  certain  degree 
of  precision,  and  hence  the  corrected  result  of  a  weighing 
may  still  be  in  error  by  an  amount  depending  on  the  preci- 
sion with  which  the  correction  itself  has  been  determined. 
Or,  again,  correcting  for  the  expansion  of  a  scale  involves  an 
experimental  investigation  of  the  coefficient  of  expansion  of 
the  material  of  which  the  scale  is  constructed,  and  this  con- 
stant can  be  determined  with  only  a  certain  degree  of  pre- 
cision. A  result  corrected  by  means  of  this  coefficient  will, 
therefore,  still  be  uncertain  beyond  a  certain  point  due  to 
the  uncertainty  in  the  value  of  the  coefficient  used.  These 
small  errors  remaining,  because  of  the  impossibility  of  com- 
pletely correcting  for  constant  errors,  are  called  residual 
errors:  then-  numerical  value  and  algebraic  sign  cannot  be 
determined,  but  usually  limiting  values  may  be  estimated 
and  assigned  to  them.  For  this  reason  they  are  properly 
grouped  and  treated  under  the  second  general  class  of  errors 
mentioned, — indeterminate  errors. 

Indeterminate  Errors. — Accidental;  Residual. —  Experience 
shows  that,  when  a  measurement  is  repeated  a  number 
of  times  with  the  same  instrument  and  by  the  same  ob- 
server under  apparently  the  same  conditions,  the  results 
usually  differ  in  the  last  place  or  sometimes  last  two 
places  of  figures.  Thus  in  so  simple  a  measurement  as  the 
determination  of  the  distance  between  two  lines  with  a  scale 
graduated  in  millimeters,  successive  measurements  will  not 
agree  to  one-tenth  millimeter  if  fractions  of  a  millimeter  are 
estimated  by  the  eye.  Errors  which  give  rise  to  such  varia- 
tions which  at  one  time  cause  a  result  to  be  too  high  and 
at  another  too  low  are  due  to  causes  over  which  the  observer 
has  no  control,  such  as  sudden  temperature  fluctuations  which 
may  give  rise  to  unequal  expansion  of  different  parts  of  an 
apparatus,  or  to  changes  in  refraction,  barometric  changes, 
shaking  of  the  instrument  due  to  mechanical  jar  or  to  the 
wind,  etc.;  and,  more  important  still,  to  physiological  causes 
arising  from  imperfections  or  fatigue  of  the  eye  or  ear 


12  PRECISION  OF  MEASUREMENTS 

of  the  observer.  The  magnitude  and  sign  of  errors  arising 
from  such  causes  have  been  shown,  however,  to  follow 
a  perfectly  definite  law, — namely,  the  law  of  chance.  The 
nature  of  this  law  may  be  illustrated  as  follows.  Suppose  a 
thousand  shots  be  fired  at  a  target  by  a  skilled  marksman 
under  conditions  as  nearly  alike  as  possible.  Experience 
shows  that  the  shots  will  be  distributed  in  a  manner  which 
at  first  sight  seems  entirely  irregular,  but  which  on  more 
careful  examination  will  be  found  to  be  approximately  in 
conformity  with  a  perfectly  definite  law.  In  an  actual  case 
obtained  with  a  target  ruled  in  horizontal  sections  by  lines 
one  foot  apart,  the  centre  line  (corresponding  to  the  bull's 
eye)  being  in  the  middle  of  one  of  these  spaces,  the  follow- 
ing results  were  obtained: — 

In  space  No.  of  shots 

1 
4 
10 
89 
190 
212 
204 
193 
79 
16 
2 

If  a  plot  be  made  with  the  number  of  shots  falling  hi  the 
several  sections  as  ordinates  and  the  distance  of  the  corre- 
sponding spaces  from  the  central  line  as  abscissae,  we  obtain 
Figure  1.  From  this  it  appears  that  plus  and  minus  devia- 
tions of  the  shots  from  the  central  line  are  about  equally 
frequent,  and  that  small  deviations  occur  with  much  greater 
frequency  than  large  ones.  If  the  number  of  shots  (corre- 
sponding to  observations)  be  increased,  the  irregularities  pres- 
ent in  the  curve  will  tend  to  smooth  out,  and  it  can  be 
shown  mathematically  that  in  the  limit  the  curve  represent- 


+  54 

to  +4* 

+  44 

"  +  34 

+  3* 

"  +  24 

+  24 

"  +14 

+  14 

"  +  4 

+  4 

"  -  4 

2 

"-14 

-14 

"  —24 

-24 

"  —  34 

-34 

"-44 

-*4 

"-54 

CURVE  OF  ERROR 


13 


ing  the  law  of  chance  takes  the  general  form  shown  in  Fig- 
ure 2,  the  equation  of  which  is 


Here  y  is  the  frequency  of  the  occurrence  of  an  error  of  the 
magnitude  z,  and  h  is  a  constant,  the  value  of  which  depends 
on  the  character  of  the  observations  and  which  affords  a 
measure  of  their  precision.  The  curve  represented  by  this 


14  PRECISION    OF   MEASUREMENTS 

equation  is  called  the  Curve  of  Error.  By  inspection  it  is 
seen  that: — 

First. — Small  errors  occur  more  frequently  than  large  ones 
(curve  of  error  has  a  maximum  for  x  =  o) ; 

Second. — Very  large -errors  are,  unlikely  to  occur  (curve  is 
asymptotic  to  the  axis  of  X) ; 

Third. — Positive  and  negative  errors  of  the  same  numerical 
magnitude  are  equally  likely  to  occur  (curve  of  error  is  sym- 
metrical with  respect  to  axis  of  Y). 

Since  accidental  and  residual  errors  of  a  series  of  observa- 
tions follow  the  law  of  chance,  they  may  be  properly  subjected 
to  mathematical  treatment  based  on  this  law.  It  must  be 
remembered,  however,  that,  since  the  law  itself  represents 
a  limiting  case,  corresponding  to  an  infinite  number  of  ob- 
servations, deductions  from  it  apply  to  a  finite  number  of 
observations  only  with  a  certain  probability  which  becomes 
less  the  smaller  the  number  of  observations. 

The  Method  of  Least  Squares. — As  already  pointed  out, 
in  the  great  majority  of  measurements  the  true  value  of  the 
quantity  is  unknown  and  cannot  be  determined.  Were  it 
known,  a  measurement  would  be  superfluous.  All  that  we 
can  hope  to  obtain  from  our  experimental  data  is  the  most 
probable  value  of  the  quantity  or  quantities  in  question. 
In  many  cases  this  is  a  simple  matter;  but  in  others,  where 
the  number  of  observations  is  larger  than  the  number  of 
unknown  quantities  to  be  determined,  the  problem  may 
become  one  of  some  difficulty.  The  branch  of  mathematics 
which  treats  of  the  general  problem  of  the  adjustment  of 
errors  of  observation  so  that  their  effect  upon  the  result  is 
reduced  to  a  minimum,  and  the  best  representative  values 
of  the  desired  quantities  thus  obtained,  is  called  Least  Squares, 
the  name  being  derived  from  the  criterion  upon  which  the 
adjustment  of  the  observations  is  based.  This  states  that 
the  most  probable  values  of  a  series  of  related  observations 
arc  those  for  which  the  sum  of  the  squares  of  the  errors  is 
a  minimum.  Certain  deductions  from  the  theory  of  Least 
Squares  will  be  assumed  as  demonstrated  in  the  course  of 


ARITHMETICAL    MEAN  15 

this  work.  For  the  proofs  the  student  is  referred  to  Bart- 
let  t's  Method  of  Least  Squares  or  other  treatises  on  the 
subject.  An  illustration  of  the  method  as  applied  to  the 
computation  of  the  constants  of  an  empirical  equation  is 
given  under  Graphical  Methods. 

The  Arithmetical  Mean.  —  Deviation  Measures.  —  We  will 
now  consider  the  precision  discussion  of  a  series  of  direct 
measurements.  Let  a1}  a2,  .  .  .  an  be  a  series  of  observations 
on  a  quantity,  all  of  which  possess  an  equal  degree  of  prob- 
ability. Under  these  conditions  the  most  probable  value 
of  the  quantity  is  given  by  the  arithmetical  mean,  m,  of  the 
series,  i.e. 


(1) 


Since  the  .true  value  of  the  quantity  is  unknown,  the  error 
of  each  of  the  observations  and  of  the  mean,  m,  cannot  be 
determined.  We  can,  however,  obtain  a  numerical  measure 
of  the  amount  by  which  each  observation  differs  from  the 
mean  value,  and  from  this  the  probable  deviation  of  the  mean 
can  be  computed.  The  difference  between  the  value  of  any 
observation  of  a  series  and  the  mean  value  of  the  series  is 
called  the  deviation  of  that  observation  from  the  mean.  It 
is  to  be  distinguished  from  the  absolute  error  of  the  obser- 
vation, i.e.,  the  difference  between  the  observed  value  and 
its  true  value,  from  which  it  may  differ  widely.  Deviations 
as  thus  computed  follow  the  same  law  as  indeterminate  errors, 
i.e.,  the  law  of  chance,  and  are  subject,  therefore,  to  the 
same  mathematical  treatment.  They  give  a  measure  of 
the  magnitude  of  the  accidental  error  of  a  measurement,  but 
evidently  afford  no  indication  of  the  presence  or  magnitude 
of  any  constant  errors  which  may  be  present. 

If  the  numerical  deviations  d^  d%,  dB,  ...  dn,  be  computed 
for  any  series  of  observations  as  above,  their  algebraic  sum 
will  be  zero,  since  the  sum  of  the  positive  deviations  is  equal 
to  the  sum  of  the  negative  deviations.  If,  however,  their 
arithmetical  mean  be  computed,  disregarding  their  sign,  the 


16  PRECISION  OF  MEASUREMENTS 

result  will  be  a  number  which  expresses  how  much  on  the 
average  any  single  observation  of  the  series  taken  at  random 
is  likely  to  differ  (plus  or  minus)  from  the  mean,  m.  This 
average  value 


n 


(2) 


is  called  the  average  deviation  of  a  single  observation,  and  will 
be  denoted  by  a.d.  In  recording  data  as  o|,  0%,  .  .  .  an,  space 
should  always  be  left  for  computing  the  deviations  di}  d%, 
.  .  .  dn,  respectively,  and  their  a.d.  as  follows:  — 

a-i  —  m  —  d1 
0  —  m  —  d 


w==—      a.d.=~ 

Looked  at  from  another  point  of  view,  an  a.d.  is  a  numerical 
measure  of  the  amount  by  which  a  new  observation  taken 
under  the  same  conditions  as  before  is  likely  to  differ  from 
the  mean  value,  m.  It  gives  a  numerical  measure  of  the 
reliability  of  any  single  observation  of  the  series  so  far 
as  accidental  errors  affecting  the  measurement  are  con- 
cerned. 

Deviation  of  the  Mean,  A.D. — In  general,  however,  it  is 
the  reliability  of  the  mean  that  we  desire  to  know  rather 
than  that  of  a  single  observation.  As  the  mean  has  a  higher 
degree  of  probability  than  any  single  observation  from  which 
it  is  computed,  it  must  evidently  have  a  smaller  deviation 
than  a  single  observation  in  proportion  to  its  greater  reli- 
ability. It  can  be  shown  that  an  arithmetical  mean  com- 
puted from  n  equally  probable  observations  is  Vn  times  as 
reliable  as  any  one  observation.  Hence,  if  the  deviation 


DEVIATION   MEASURES  17 


measure  of  a  single  observation  of  a  series  is  a.d.,  the  devia- 

tion measure  of  the  mean  of  n  such  observation  is  only—  — 

Vn 

as  great;  i.e.,  the  deviation  of  the  mean,  denoted  by  A.D.,  is, 


Thus,  if  the  mean  value  of  nine  measurements  of  the  dis- 
tance between  two  lines  is  1.3215  mm.  and  the  average  devia- 
tion of  any  one  of  the  measurements  is  found  to  be  a.d.  = 
0.0033  mm.,  the  mean  will  have  a  probable  deviation  not 

0.0033 
greater  than  —7=  —  =  0.0011  mm.    From  this  it  will  be  seen 

that  in  general  it  does  not  pay  to  increase  the  number  of 
observations  beyond  a  certain  limit,  say  nine  or  sixteen,  as  the 
time  and  labor  involved  soon  become  excessive,  without  a 
corresponding  increase  in  the  precision  attained. 

Fractional  and  Percentage  Deviation  Measures.  —  It  is  fre- 
quently convenient  to  express  the  reliability  of  a  quan- 
tity as  a  fractional  or  as  a  percentage  part  of  the  quantity 
itself.  Thus  we  have  in  very  common  use  the  two  following 
deviation  measures  derived  from  the  preceding:  — 

the  fractional  deviation  of  a  single  observation  =  ^-i  • 

the  percentage  deviation  of  a  single  observation  =  100  —  '-  • 

a  ' 

A  D 

the  fractional  deviation  of  the  mean  =  —  -  —  -  • 

m     ' 

A  D 

the  percentage  deviation  of  the  mean  =  100  -  —  '-. 

in 

Since  these  measures  are  never  computed  to  more  than  two 
significant  figures,  see  page  23,  a  and  m,  being  approxi- 
mately the  same,  may  be  used  indiscriminately  in  the  com- 
putation. 

Deviation  Measure  vs.  Precision  Measure.  —  A  little  con- 
sideration will  make  clear  that  all  of  the  above  deviation 
measures  give  a  measure  of  the  magnitude  of  errors  of  that 


18  PRECISION   OF   MEASUREMENTS 

type  which  has  been  classed  as  accidental.  A  result  may 
be  seriously  in  error  due  to  residual  errors,  and  yet  the  ob- 
servations show  a  good  agreement  among  themselves,  and 
their  deviation  measure  be  correspondingly  small.  If  the 
magnitude  of  the  residual  errors  can  be  estimated,  we  may 
compute  the  true  precision  measure,  abbreviated  p.m.,  of 
the  result  as  follows: — 

Let  the  estimated  magnitude  of  the  residual  errors  be 
r\j  r2)  •  •  -  rn>  Let  d.m.  represent  the  value  of  the  deviation 
measure  of  the  accidental  errors.  This  may  be  expressed  as 
an  average,  fractional  or  percentage  deviation,  but,  which- 
ever is  chosen,  the  residuals  must  be  expressed  in  the  same 
way.  It  can  then  be  shown  that  the  most  probable  measure 
of  the  reliability  of  the  result  will  be  given  by  the  expression : — 


p.m.2  =  d.m.2  +  rf  +  r*  +  .  .  .  +f**,  (4) 


or         p.m.  =  -yd.ra.2  +  r\  +  r22  +  .  .  .  +  rn2.        (4a) 

Thus  the  precision  measure  of  a  result  differs  from  its  devia- 
tion measure  in  that  it  includes  the  effect  of  residual  as  well 
as  of  accidental  errors.  In  a  great  many  cases  the  value  of 
the  residuals  is  negligible  compared  with  the  magnitude  of 
the  accidental  errors.  In  this  case  p.m.  =  d.m.  The  synir 
bol  8  will  be  used  to  represent  the  value  of  p.m.  or  d.m.  in- 
discriminately, as  the  latter  is  only  a  special  case  of  the 
former  when  Sr2  is  negligible. 

It  will  be  shown  on  page  31  that  any  single  residual  rk 
may  be  regarded  as  negligible  in  computing  p.m.  if 

rt  =  J  p.m.  (5) 

Also  that  any  number  p  of  residuals  are  simultaneously  neg- 
ligible if 

Vri'  +  r,»  +  .  .  .  +  V  =  J  p.m.  (6) 

The  Probable  Error  and  the  Mean  Error.  —  In  the  discus- 
sion of  observations  by  the  method  of  Least  Squares  and 
in  many  foreign  treatises  certain  other  measures  are  in 
common  use;  namely,  the  so-called  probable  error  and  the 


PROBABLE  ERROR  AND  MEAN  ERROR  19 

mean  error.  The  "probable  error"  of  an  observation  is  of 
such  a  magnitude  that  the  probability  of  making  an  error 
greater  than  it  is  just  equal  to  the  probability  of  making 
one  less  than  it,  both  probabilities  being  one-half.  The 
probable  error  of  a  single  observation  and  of  the  mean 
of  n  observations  are  given  by  the  expressions 

p.e.  =  0.6745  J   ^    and  P.E.  =  0.6745  \J  __  ^  - 

'  n  —  1  r  n  (n  —  1) 

respectively,  where  3d2  is  the  sum  of  the  squares  of  the 
deviations  of  the  single  observations  from  the  mean. 
The  following  approximate  formulae  are  more  convenient 
forms  to  use  for  purposes  of  computation: 

p.e.  =  0.8453  ,—  J^=  and  P.E.  =  0.8453    JJ—  .  • 

\n  (n  —  1)  n\n  —  1 

The  "mean  error"  ^  is  defined  as  the  square  root  of  the 
arithmetical  mean  of  the  squares  of  the  errors.  It  is 
seldom  used  except  in  treatises  on  Least  Squares. 

It  can  be  shown  from  the  equation  of  the  curve  of  error 
(p.  13)  that,  interpreted  geometrically,  p.e.  =  OP,  the  ab- 
scissa of  the  ordinate  which  divides  the  area  OXY  into 
equal  parts;  a.d.  =OD,  the  abscissa  of  the  ordinate  pass- 
ing through  the  center  of  gravity  of  the  half  area;  and 
p=OM,  the  abscissa  of  the  point  of  inflection  of  the  curve. 
From  this  it  follows  that 

0.4769  1  1 

P.e.=  --;    «.<*. 


or  p.e.  =  0.85  a.d.  =  0.67/*. 

Although  the  probable  error  is  frequently  used  by  physi- 
cists as  a/  precision  measure,  the  average  deviation  is 
simpler,  and  will  be  adopted  throughout  the  present  work. 
Weights.  —  It  frequently  happens  that  it  is  necessary  to 
average  a  series  of  results  which  have  not  been  taken  under 
like  conditions,  and  which  are  not  all  equally  probable  ;  i.e., 


20  PRECISION   OF  MEASUREMENTS 

which  do  not  have  the  same  precision  measures.  In  this  case 
it  is  first  necessary  to  assign  relative  weights  to  the  obser- 
vations, so  that,  in  taking  the  average,  the  more  precise 
measurements  may  be  given  a  proportionally  greater 
"  weight "  than  the  less  precise  measurements. 

It  can  be  shown  that  the  relative  weights  of  a  series  of 
observations  are  inversely  proportional  to  the  squares  of 
their  precision  measures;  i.e.,  if  p1;  pz,  p3,  .  .  .  etc.,  are  the 
weights  of  a  series  of  observations  whose  respective  precision 
measures  are  Bl9  S2,  $3,  •  •  •  etc.,  respectively, 

1  .  i  .  i  .  ,« 

PI  •  P2  •  PS  •  •  •  •  =:==^  •  7T2  •  jr| l*J 

In  determining  the  values  of  p}  the  nearest  round  numbers 
satisfying  the  above  proportion  should  be  chosen. 

Since  the  various  precision  and  deviation  measures  differ 
from  each  other  only  by  a  constant  factor,  any  one  of  them 
may  be  used  in  computing  "  weights."  It  is,  of  course,  neces- 
sary, however,  that  the  same  measure  be  used  throughout 
any  given  discussion;  i.e.,  it  is  not  permissible  to  express 
the  precision  of  one  quantity  as  an  average  deviation,  another 
as  a  probable  error,  and  a  third  as  a  percentage  error. 

The  Weighted  Mean. — Having  obtained  the  weights  pi,  p2, 
p8)  etc.,  to  be  assigned  respectively  to  a  series  of  quantities 
Wj,  w2,  ms,  etc.,  the  best  representative  value  or  weighted 
mean  will  evidently  be  given  by  the  expression 

ViXm-i  +  VvX  m»  +  p«  X  m*  ...  /ox 

—  •  (8) 


Pi  +  Pz  +  Ps  +  •  •  - 

Rejection  of  Observations. — In  a  series  of  measurements 
taken  under  similar  conditions,  it  not  unfrequently  happens 
that  an  observation  will  differ  quite  widely  from  others  in 
the  series,  and  the  tendency  to  regard  such  an  observation 
as  erroneous  and  to  reject  it  is  great,  particularly  among 
beginners.  If  such  an  observation  obviously  contains  a 
mistake,  as,  for  example,  the  recording  of  a  wrong  number, 
the  recording  the  wrong  scale  division,  the  incorrect  adding 


REJECTION  OF   OBSERVATIONS  21 

up  of  weights,  etc.,  it  may,  of  course,  be  legitimately  re- 
jected. If,  however,  no  mistake  is  apparent,  the  observa- 
tion should  never  be  rejected  without  the  most  scrupu- 
lously unbiassed  judgment  on  the  part  of  the  observer  or 
the  application  of  some  mathematical  criterion  for  the 
rejection  of  doubtful  observations.  For  the  experienced 
observer  the  former  procedure  is  preferable,  even  though 
several  mathematical  criteria,  Peirce's,  Chauvenet's,  etc., 
have  been  deduced,  which  are  very  satisfactory  when 
the  number  of  observations  considered  is  large.  In  most 
physical  work  the  number  of  observations  is  not  very 
great,  however,  and  one  widely  discordant  from  the  others 
has  an  undue  weight  on  the  value  of  the  mean.  It  is 
frequently  better  to  reject  such  an  observation,  even 
though  it  contains  no  apparent  mistake.  A  good  cri- 
terion to  follow  in  such  cases  is  the  following: — 

Compute  the  mean  and  the  average  deviation  a.d., 
omitting  the  doubtful  observation.  Compute  also  the 
deviation,  d,  of  the  doubtful  observation  from  the  mean. 
If  d  >T  4  a.d.,  reject  the  observation,  since  it  can  be  shown 
that  the  probability  of  the  occurrence  of  an  observation 
whose  deviation  is  equal  to  four  times  the  average  devia- 
tion is  only  one  in  a  thousand.  An  error  of  this  unusual 
magnitude  is  called  a  Huge  Error. 

Computation  Rules  and  Significant  Figures. — It  is  prob- 
ably true  that  at  least  half  the  time  usually  spent  on  com- 
putations is  wasted,  owing  to  the  retention  of  more  figures 
than  the  precision  of  the  data  warrants,  and  to  the  failure 
to  use  either  logarithms  or  a  slide  rule  instead  of  the 
lengthy  arithmetical  processes  of  multiplication  and  divi- 
sion. An  important  feature  of  physical  laboratory  work  is 
the  proper  use  of  significant  figures  in  recording  data  and 
in  subsequent  computations.  The  habit  should  be  ac- 
quired at  the  outset  of  rejecting  at  each  stage  of  the  work  all 
figures  which  have  no  influence  on  the  final  result. 

Rules  for  the  correct  use  of  significant  figures  are  dis- 
cussed in  the  introduction  to  Holman's  "Computation 


22  PRECISION   OF   MEASUREMENTS 

Rules  and  Logarithm  Tables"  which  may  very  advantageously 
be  used  in  connection  with  the  laboratory  work.  A  fuller 
discussion,  including  the  demonstration  of  these  rules,  is 
given  in  Holman's  "Precision  of  Measurements."  The 
following  brief  statement  of  the  rules  is  essentially  that 
given  in  these  works: — 

A  Digit  is  any  one  of  the  ten  characters  1,  2,  3,  4,  5,  6,  7,  8, 
9,0. 

A  Significant  Figure  is  any  digit  to  denote  or  signify  the 
amount  of  the  quantity  in  the  place  in  which  it  stands.  Thus 
zero  may  be  a  significant  figure  when  it  is  written,  not  merely 
to  locate  the  decimal  point,  but  to  indicate  that  the  quantity 
in  the  place  in  which  it  stands  is  known  to  be  nearer  to  zero 
than  to  any  other  digit. 

For  example,  if  a  distance  has  been  measured  to  the  nearest 
fiftieth  of  an  inch,  and  found  to  be  205.46  inches,  all  five  of 
the  figures,  including  the  zero,  are  significant.  Similarly,  if 
the  measurement  had  shown  the  distance  to  be  nearer  to 
205.40  than  to  205.41  or  to  205.39,  the  final  zero  would  be 
also  significant,  and  should  invariably  be  retained,  since  its 
presence  serves  the  most  useful  purpose  of  showing  that  this 
place  of  figures  had  been  measured  as  well  as  the  rest.  If 
in  such  a  case  the  quantity  had  been  written  205.4  instead 
of  205.40,  the  inference  would  be  drawn  either  that  the 
hundredths  of  an  inch  had  not  been  measured  or  that 
the  person  who  wrote  the  number  was  ignorant  or  careless 
of  the  proper  numerical  usage.  Failure  to  follow  this 
simple  rule  is  a  common  source  of  annoyance  and  un- 
certainty. 

A  zero,  when  used  merely  to  locate  the  decimal  point,  is 
not  a  significant  figure  in  the  above  sense;  for  the  position 
of  the  decimal  point  in  any  measurement  is  determined  solely 
by  the  unit  in  which  the  quantity  in  question  is  expressed. 
The  number  of  decimal  places  in  a  result  has,  therefore,  in 
itself  no  significance  in  indicating  the  precision  of  a  measure- 
ment. For  example,  suppose  a  certain  distance  is  found  to 
be  122.48  cm.  with  a  A.D.  of  0.12  cm.  The  percentage  pre- 


RULES   FOR   SIGNIFICANT   FIGURES  23 

0  12 
cision  of  the  measurement  is  -J—  -  X  100  =  0.10%.    The  re- 


suit  contains  five  significant  figures,  and  its  precision  remains 
the  same,  namely,  0.10%,  whether  it  be  expressed  as  1.2248 
meters,  A.D.  =  0.0012  m.,  or  1224.8  mm.,  A.D.  —  1.2  mm. 
The  statement  that  the  distance  is  measured  to  0.12  cm.  gives 
no  idea  of  the  precision  of  the  measurement  unless  the  distance 
itself  is  stated.  A  fractional  or  percentage  precision  measure, 
on  the  other  hand,  gives  a  definite  idea  of  the  precision  of  the 
measurement  without  any  further  statement,  as  it  involves 
both  the  value  of  the  quantity  and  its  average  deviation. 

The  following  rules  are  deduced  subject  to  the  condition 
that  the  accumulated  errors  in  a  computation  shall  not  affect 
the  second  unreliable  place  of  figures  in  the  final  result  by 
more  than  one  unit,  even  though  as  many  as  sixteen  rejec- 
tions of  figures  are  made  in  the  course  of  the  computation. 
This  is  a  safe  limit  to  assume  for  most  physical  work,  as  it 
is  seldom  that  more  than  this  number  of  quantities  or  oper- 
ations enter  into  any  single  computation. 

Rule  I.  —  In  rejecting  superfluous  figures,  increase  by  1  the 
last  figure  retained,  if  the  following  figure  (that  rejected)  is  5 
or  over. 

Rule  II.—  In  all  deviation  and  precision  measures  retain 
two,  and  only  two,  significant  figures. 

The  reason  for  this  rule  is  as  follows:  consider  the  above 
example  where  the  length  measured  is  m  =  122.48  cm.  with  an 
A.D.  —  0.12  cm.  The  significance  of  the  A.D.  is  that  the 
place  of  figures  in  m  occupied  by  the  4  is  uncertain  by  1 
unit,  and  that  the  next  place  of  figures  occupied  by  8  is  un- 
certain by  12  units,  while  the  third  decimal  place  would  be 
uncertain  by  at  least  120  units;  i.e.,  by  an  amount  which 
would  render  it  practically  worthless.  In  general,  the  place 
of  figures  corresponding  to  the  first  significant  figure  of  the 
deviation  measure  is  somewhat  uncertain  (from  1  to  9  units), 
while  the  place  corresponding  to  the  second  significant  figure 
in  the  deviation  measure  is  uncertain  by  ten  times  this  amount 
(10  to  90  units,  or,  more  exactly,  10  to  99  units).  Beyond 


24  PRECISION   OF  MEASUREMENTS 

this  place  the  significance  of  additional  figures  is  so  slight  as 
to  be  of  no  value :  hence,  as  deviations  and  precision  meas- 
ures are  at  best  only  estimates  of  the  reliability  of  a  result, 
it  is  useless  to  compute  them  to  places  of  figures  which  have 
no  real  significance  in  the  result  to  which  they  refer. 

If  the  first  significant  figure  of  the  precision  measure  is  as 
great  as  8  or  9,  in  which  case  the  place  of  figures  in  the  data 
corresponding  to  the  second  place  in  the  precision  measure 
is  unreliable  by  80  to  90  units,  it  is  usually  sufficient  to 
retain  but  one  significant  figure  in  the  precision  measure. 

Rule  III. — Retain  as  many  places  of  figures  in  a  mean  re- 
sult and  in  data  in  general  as  correspond  to  the  second  place 
of  significant  figures  in  the  deviation  or  precision  measure. 

Two  places  of  doubtful  figures  are  thus  retained  in  data 
and  computations  rather  than  one,  so  that  accumulated 
errors  due  to  rejections  in  the  course  of  a  computation  may 
not  affect  the  first  place  of  uncertain  figures  in  the  result. 

Rule  IV. — The  sum  or  difference  of  two  or  more  quanti- 
ties cannot  be  more  precise  numerically  than  the  quantity 
having  the  largest  average  deviation.  Hence,  in  adding  or 
subtracting  a  number  of  quantities,  find  the  average  deviation 
of  each,  and  then  retain  in  each  quantity  as  many  places  of 
figures  as  correspond  to  the  second  place  of  significant  figures 
in  the  largest  deviation. 

Rule  V. — In  multiplication  or  division  the  percentage 
precision  of  the  product  or  quotien  cannot  be  greater  than 
the  percentage  precision  of  the  least  precise  factor  entering 
into  the  computation.  Hence,  in  computations  involving 
these  operations,  the  number  of  significant  figures  to  be  re- 
tained in  each  factor  is  determined  by  the  number  properly 
retained  under  Rule  III.  in  the  factor  which  has  the  largest 
percentage  deviation.  Computations  involving  a  precision 
not  greater  than  ^  Per  cent,  should  be  made  with  a  slide 
rule.  For  greater  precision  logarithm  tables  should  be 
used.  If  multiplication  and  division  must  be  resorted  to, 
the  "short  method"  of  rejecting  all  superfluous  figures 
at  each  stage  of  the  operation  should  be  adopted. 


INDIRECT  MEASUREMENTS  25 

Rule  VI. — In  carrying  out  the  operations  of  multiplication 
and  division  by  logarithms,  retain  as  many  figures  in  the  man- 
tissa of  the  logarithm  of  each  factor  as  are  properly  retained 
in  the  factors  themselves  under  Rule  V. 

Precision  Discussion  of  Indirect  Measurements. — We  will 
now  consider  the  precision  discussion  of  indirect  measure- 
ments; i.e.,  those  in  which  the  final  result  is  a  more  or  less 
complicated  function  of  one  or  more  directly  measured 
quantities.  Two  distinct  classes  of  problems  may  arise: 

First. — The  precision  measures  of  the  directly  measured 
components  are  known  (determined  as  above  described),  and 
it  is  desired  to  find  the  precision  measure  of  the  final  result. 

Second. — The  desired  precision  of  the  final  result  is  stipu- 
lated at  the  outset,  and  the  problem  is  to  ascertain  what 
precision  is  necessary  in  the  components,  in  order  that  the 
accumulated  effect  of  the  errors  in  these  on  the  final  result 
shall  not  exceed  the  prescribed  limit. 

The  importance  of  these  problems  cannot  be  overesti- 
mated; for,  in  the  first  case,  a  final  result,  be  it  the  result 
of  chemical  analysis,  the  value  of  a  physical  constant,  the 
algebraic  expression  of  a  law,  or  an  efficiency  test  of  an  en- 
gine, is  practically  worthless  unless  a  numerical  estimate 
of  its  reliability  can  be  stated.  In  fact,  it  may  be  worse 
than  worthless  if  carried  out  to  indicate  a  higher  precision 
than  the  data  warrant.  And  the  second  case  is  of  equal 
importance;  for,  unless  an  investigator  makes  a  preliminary 
precision  discussion  of  his  method  and  apparatus  before 
beginning  work,  so  that  he  may  know,  at  least  approx- 
imately, how  precisely  each  quantity  entering  into  the  final 
result  should  be  measured,  the  chances  are  that  much 
time  and  labor  will  be  wasted  in  measuring  some  compo- 
nents more  precisely  than  necessary,  while  others  will  be 
measured  to  a  degree  of  precision  which  will  render  impos- 
sible the  attainment  of  the  desired  precision  in  the  final 
result. 

Notation. — In  the  precision  discussion  which  follows,  the 
following  notation  will  be  adopted. 


26  PRECISION   OF   MEASUREMENTS 

M  =  the  final  computed  result  of  any  indirectly  measured 
quantity. 

A  =  numerical  precision  measure  of  M. 

mij  mz,  .  .  .  =  directly  measured  quantities,  which  may 
be  either  mean  results  or  single  observations. 

Si,  52,...  =  the  numerical  precision  measures  of  m1}  mz, 
.  .  .  respectively. 

The  values  of  5  might  be  expressed  as  average  deviations, 
probable  errors,  or  mean  errors,  discussed  on  page  19.  In  the 
discussion  of  any  given  problem,  however,  the  same  kind 
of  precision  measure  must  be  used  throughout;  i.e.,  in  any 
given  problem  it  is  not  proper  to  express  the  precision  meas- 
ures of  some  quantities  as  probable  errors,  others  as  average 
deviations,  and  still  others  as  percentage  or  fractional  de- 
viations. In  the  following  discussion  we  shall  always  assume 
values  of  8  to  be  expressed  as  deviations. 

A1;  A2,  .  .  .  ,  will  be  used  to  denote  the  deviations  pro- 
duced in  M  by  deviations  Si,  82>  •  •  •  m  the  components 
%,  ra2,  .  .  .  respectively. 

From  the  above  notation  it  follows  that, 

—  —  the  fractional  precision  of  the  final  result ; 

100  —  =  percentage  precision  of  the  final  result ; 

a       $ 

—  ,—  ,...=  the  fractional  precision  of   the   components 

7?ll      m^ 

mi>  mz>  -  •  -  respectively; 

OJ  $ 

100  —  ,  100  —  .       .  =  the  percentage  precision  of  the  com- 
rax  m% 

ponents  ml}  w2,  .  .  .  respectively. 
In  general 

M  =  f  (mlt  ?n3,  .  .  .  mn),  (9) 

which  for  brevity  may  be  written  M  =  f  (  ),  where  the 
form  of  the  function  is  determined  by  the  formula  by  which 
M  is  computed  from  raj,  ra2,  etc.  The  first  class  of  problems 
may  then  be  stated  mathematically  as  follows : — 


INDIRECT   MEASUREMENTS  27 

Case  I.  —  The  Direct  Problem.  Given  the  precision  meas- 
ures 81?  S2,  .  .  .  8n,  of  the  component  measurements  ml}  m2, 
.  .  .  mn,  to  compute  the  precision  measure  A  of  the  result  M. 

The  solution  of  this  problem  is  obtained  by  finding,  first, 
the  effect  of  the  deviation  in  each  component  on  M,  and 
then  combining  these  separate  effects  to  get  the  resultant 
effect.  The  method  of  computation  to  be  followed  in  this 
last  procedure  depends  upon  the  law  to  which  the  deviations 
concerned  are  subject,  and  will  be  considered  below. 

Separate  Effects.  —  The  effect  of  a  deviation  8*  in  any  com- 
ponent mk  will  be  to  produce  a  deviation  A*  in  M  of  an 
amount 


i.e.,  an  amount  equal  to  the  rate  at  which  the  value  of  the 
function  M  =  f(  )  changes,  as  mk  changes  (the  other  com- 
ponents m2,  ra3,  .  .  .  etc.,  remaining  constant),  multiplied 
by  the  actual  change  8#  in  mk,  or,  in  other  words,  the  partial 
differential  coefficient  of  the  function  with  respect  to  m* 
multiplied  by  the  actual  deviation  8*  in  mt. 

Example  1.  —  Find  the  deviation  in  the  volume  of  a 
sphere  whose  diameter  is  10.013  cm.,  if  the  average  devia- 
tion in  the  measurement  of  the  diameter  is  A.D.  —  0.012  cm. 

V  =  ±*D*. 

Comparing  with  the  notation  on  page  26,  it  is  evident  that 
M=V  =  f(     )  =  %TrD*, 

m  =  D  =  10.013  cm.,  and  5  =  A.D.  =  0.012  cm. 
The  computed  value  of  V  is 

V  =  i  X  3.1416  X  10.013* 

=  525.52  c.c. 

By  (10)  the  deviation  A  in  this  volume  produced  by  the 
deviation  5  in  the  diameter  is 

A=Aa^3).5 

=  i  IT.  3D2  .  5 

=  i  X  3.1  X  3  X  102      0.012 

=  1.9  c.c.  ; 


28  PRECISION   OF  MEASUREMENTS 


i.e.,  the  volume  525.5  c.c.  is  uncertain  by  1 .9  c.c.,  or  by  19  parts 

0.012 
D  ~  ™     10 


in  5300.     A  deviation  of  100  ^  =  100  ^^  =  0.12%  in  the 


A  19 

diameter  introduces  a  deviation  of  100  y  —  100  -^  —  0.36% 

in  the  volume,  i.e.,  a  percentage  deviation  three  times 
as  great.  In  this  case  V  is  a  function  of  only  a  single  vari- 
able, hence  the  resultant  deviation  in  V  is  given  at  once  by 
the  above  result. 

Example  2. — What  will  be  the  numerical  deviation  in  the 
value  of  g,  as  determined  by  a  second's  pendulum,  due  to 
a  deviation  A.D.  =  0.0020  second  in  the  determination  of 
the  time  of  vibration,  and  a  deviation  A.D.  =  0.10  cm.  in 
the  determination  of  the  length? 


9  = 


Hence    in    the     general    notation   M  =  g  =  f  (    )  =  —^  • 

mi=l=WO    cm.;    mz  =  t  —  1   sec.;    5i  =  5i  =  0.10  cm.g 
52  =  Si  =  0.0020  sec. 

The  deviation  Aj  in  g  produced  by  the  deviation  Si  in  I  is 
by  (10) 


=  ^~  X  1  X  0.10 


sec. 

i.e.,  a  deviation  of  0.10  cm.  in  the  measurement  of  I  will 

cm 
produce  a  deviation  of  0.96  —  -2  in  the  value  of  g. 

Similarly  the  deviation  At  in  g  due  to  the  deviation  8t  in  t 
is  by  (10) 


HfcS) 


2  X3-1128X1°°X  0.0020 

cm.  « 


=  —3.8 

sec. 


RESULTANT   EFFECT  29 

i.e.,  a  deviation  of  0.0020  sec.  in  the  measurement  of  the  time 

cm. 

will  introduce  an  uncertainty  in  the  value  of  g  of  3.8  ==«  • 

sec. 

The  negative  sign  simply  indicates  that  a  positive  deviation  in 
t  produces  a  negative  deviation  in  g,  and  vice  versa.  Since  all 
deviations  are  equally  likely  to  be  plus  or  minus,  in  pre- 
cision discussions  no  attention  is  usually  paid  to  the  sign 
resulting  from  differentiation  of  a  function.  By  a  direct 
application  of  (10)  the  effect  of  a  deviation  in  any  single 
component  on  a  final  result  may  always  be  computed.  A 
much  shorter  method  than  the  above,  applicable  in  certain 
special  cases,  will  be  pointed  out  below. 

Resultant  Effect.  —  To  find  the  combined  or  resultant  effect  A, 
on  the  final  result  of  the  separate  deviations  AJ,  A2,  .  .  .  etc., 
produced  by  the  components. 

If  for  any  reason  the  values  of  Ab  A2,  ...  etc.,  are  of 
specified  magnitude  and  sign  (in  which  case  they  would  not 
follow  the  general  law  of  deviations),  they  should  be  com- 
bined according  to  the  formula 

A  =  A!  +  A2  +  .  .  .  +  An.  (11) 

As  this  case  rarely  occurs  in  practice,  it  need  not  be  further 
discussed  here. 

The  important  case  to  consider  is  that  in  which  the  values 
of  AJ,  A2,  .  .  .  etc.,  are  equally  likely  to  be  plus  or  minus  and 
of  a  magnitude  determined  by  the  general  law  of  deviations. 
If  each  A*  is  computed  by  formula  (10),  page  27,  i.e., 


these  conditions  will  always  be  fulfilled,  since  the  values  of 
8k  which  determine  A*  are  of  the  nature  of  true  deviations. 
Under  these  circumstances  the  most  probable  resulting  devia- 
tion A,  in  M,  can  be  shown  by  the  method  of  Least  Squares  to 
be  that  obtained  by  combining  the  values  of  A*  by  the  formula 

A2  =  AX2  +  A22  +  .  .  .  +  A2n,  (12) 

or    A   =  VAl2  +  A22  +  .     .   +An2.  (12a) 


30  PRECISION   OF   MEASUREMENTS 

This  does  not  give  us  an  exact  solution  of  the  problem,  but 
rather  the  solution  which  in  the  long  run  is  better  than  that 
obtained  by  any  other  method  of  combining  the  values  of 
A*.  It  is  to  be  noted  that  by  this  method  of  computation 
the  effect  of  the  sign  of  individual  deviations  A*  is  elim- 
inated. The  resultant  deviation  A  is,  of  course,  to  be  re- 
garded as  equally  likely  plus  or  minus. 

Example  2  (continued). —  Thus  in  Example  2  the  com- 
bined effect  of  the  deviations  in  I  and  in  t  on  the  value  of  g 
is  to  be  found  by  taking  the  square  root  of  the  sum  of  the 
squares  of  the  deviations  Aj  and  &t,  which  61  and  <$t  sep- 
arately produce  in  g  respectively;  i.e., 


•  * 


—  Vo.962  +  3.82 

cm. 

=  3.9=^ 
sec. 

Hence  a  deviation  of  0.10  cm.  in  Z  and  0.0020  sec.  in  t  will 

make  the  value  of  g  =  980  ===%  uncertain  by  nearly  4  ==3. 
sec.  sec. 


iterion  for  Negligibility  of  Deviations  in  Components. — It 

is  frequently  important  to  determine  whether  the  deviation 
arising  from  one  or  more  components  may  be  neglected 
in  computing  the  A  of  the  final  result.  For  this  purpose 
the  following  criterion  may  be  deduced. 

As  explained  under  rules  for  significant  figures  on  page 
23,  two  significant  figures  are  all  that  should  be  retained 
in  any  deviation  measure.  A  quantity  which  affects  a  re- 
sult by  only  yg-  the  amount  of  its  deviation  or  precision 
measure  will  therefore  affect  it  only  in  that  place  of  significant 
figures  corresponding  to  the  second  place  in  the  deviation 
measure.  This  place  is  so  uncertain  that  such  an  amount 
may  in  general  be  regarded  as  negligible.  Although  the 
assumption  that  •£$  p.m.  or  -fa  d.m.  is  negligible  is  some- 
what arbitrary,  it  has  been  found  to  be  a  convenient  and 
practical  criterion  to  adopt. 

Suppose,  therefore,  that  the  value  of  the  A  of  some  quantity 


NEGLIGIBILITY   OF   DEVIATIONS  31 

M  is  made  up  of  deviations  Ax,  A2,  .  .  .  An,  arising  from 
various  deviations  S1;  82>  •  •  •  Sn,  in  components  mlf  mz,  .  .  . 
mn.  May  any  of  these  A's,  as  A*,  be  neglected  in  com- 
puting A,  or,  in  other  words,  may  any  of  the  components, 
as  mt,  be  regarded  as  being  without  sensible  error  on  Mf 
To  answer  this  question,  let 

-fA22+.      .+A,2+...An2 


and  A'  =  V  Ax2  +  A22  -f  .  .  .  +  An2  with  A*  omitted. 
Then,  if         A  —  A'    <  TV  A 

or  A'  >  0.9  A, 

by  the  above  criterion  A*  may  be  considered  as  negligible. 
But  A/  =  A2  —  A'2 

=  A2(12  —  0.92) 
=  0.19  A2 
/.  A*  =  0.43  A. 

Hence  the  deviation  in  any  component  m  may  be  neglected 
in  computing  the  A  of  M  ,  if  its  effect  on  M  is  equal  or  less 
than  0.43  A.  A  still  safer  and  more  convenient  criterion  to 
adopt,  since  the  number  of  components  considered  is  usually 
small  and  hence  the  assumed  formula  of  squares  is  less 
rigidly  applicable,  is 

A*  ^  0.33  A  ^1  A.  (13) 

In  the  same  way  it  can  be  shown  that  deviations  in  any 
number,  p,  components  are  simultaneously  negligible  if 

VAl2  +  A22  +  ...Ap2  <  J  A.  (14) 

The  above  criterion  also  applies  to  the  rejection  of  residuals 
in  computing  the  value  of  the  precision  measure  by  the 


formula  p.m.  =     d.m.2  +  r-f  +  r22  +   .  .   .  rn2,    as    stated 
on  page  18. 

Case  II.  —  The  Converse  Problem.  Given  a  prescribed  pre- 
cision A  to  be  attained  in  the  final  result  M,  to  find  the  allow- 
able deviations  §1}  82>  &£••>  in  the  components  ml}  w2,  etc.,  re- 


32  PRECISION   OF  MEASUREMENTS 

spectively,  such  that  their  combined  effect  on  M  shall  not  exceed 
the  value  of  A. 

We  have  seen  that  when  the  deviations   follow  the   law 
of  errors, 


If  the  value  of  A  is  given  and  no  further  conditions  im- 
posed, there  are  evidently  an  infinite  number  of  solutions 
to  the  problem;  i.e.,  an  indefinite  number  of  values  can  be 
found  for  Ab  A2,  .  .  .  etc.  (and  hence  for  the  correspond- 
ing values  of  8^  S2>  •  •  •  etc.),  which  will  satisfy  the  above 
equation. 

The  most  advantageous  distribution  of  errors  among  the 
components  will  evidently  be  that  one  by  which  the  desired 
precision  is  obtained  with  the  minimum  expenditure  of 
time  and  labor  on  the  part  of  the  experimenter.  As  this 
will  vary  greatly  with  each  individual  problem,  no  mathe- 
matical criterion  can  be  formulated  which  will  embrace 
all  cases.  It  is  best,  therefore,  at  least  for  a  preliminary 
distribution  of  errors  among  the  components,  to  so  adjust 
them  that  the  errors  inherent  hi  each  variable  or  component 
shall  produce  the  same  effect  on  the  final  result.  This  is 
spoken  of  as  the  solution  of  the  problem  for  "equal  effects." 

Solving  the  formula  for  resultant  effects  subject  to  this 
condition,  i.e., 


=  A,"  _ 


we  have 

* 

hence  for  any  component, 


(16) 


Having  thus  determined  A*,  the  corresponding  value  of 
can  be  found  at  once  by  equation  (15). 


EQUAL   EFFECTS  33 

Example  3. — How  precisely  should  the  time  of  vibration 
and  length  of  a  seconds  pendulum  be  measured  in  order  that 
the  computed  value  of  g  may  be  reliable  to  one-tenth  of  one 

percent.?     g=  ^-- 

As  the  pendulum  is  stated  to  be  a  "seconds"  pendulum, 
t  =  one  second  and  Z  =  100  cm.  approximately.  It  is  further 

stipulated  that  100—  ^0.10;  hence  the  allowable  resultant 

deviation  A  in  g  must  not  be  greater  than  A  =  g  x  0.0010 

cm. 

=  0.98  =4      We  are  to  find  the  allowable  values  of  8t  and  Si, 
sec. 

which  will  give  this  precision.  Solving  the  problem  subject 
to  the  condition  of  equal  effects, — -i.e.,  that  the  resultant  de- 
viation in  g  is  caused  equally  by  the  deviation  in  t  and  in  I, — 

we  have 

A       0.98  cm.  cm. 

A,  =  A*  =  -j=  =  — ^-  =2  =  0.70=^  . 
yn       y2  sec.  sec. 

Hence  the  allowable  deviations  in  the  time  and  length  meas- 
urements must  be  reduced  to  such  a  magnitude  that  they  do 

not  separately  produce  a  deviation  in  g  greater  than  0.70 ^ 

sec. 

respectively.     But  by  the  general  equation  (10),  page  27 


7_  cm.      2  x  3. 12  X  100  cm. 

or  0.70  =3  =  —     —  —    -  .  St 

sec.  1  sec. 

hence  5t  =  0.00037  sec. 
Shnilarly,       «-*.•,- 

cm.          3 

or  0.70  =r  =  —=2  •  Si 

sec.        I2  sec. 

hence  81  =  0.073  cm. 

The  time  of  vibration  of  the  pendulum  should  there- 
fore be  measured  to  0.00037  sec.,  and  its  length  measured  to 
0.073  cm. 

The  Fractional  or  Percentage  Method  of  Solution. — The 
preceding  formulae  for  obtaining  the  precision  of  a  final 
result  from  the  known  or  estimated  precision  of  the  com- 


34  PRECISION   OF  MEASUREMENTS 

ponent  measurements,  and  for  calculating  the  necessary 
precision  of  the  component  measurements  when  the  de- 
sired precision  of  the  final  result  is  stipulated,  are  entirely 
general,  and  by  them  any  type  of  problem  can  be  solved. 
It  is  to  be  particularly  noted  throughout  the  preceding 
discussion  that  the  values  of  the  precision  or  deviation 
measures  8  and  A  are  numerical  deviations  expressed  in 
the  same  units  as  the  quantities  to  which  they  refer. 
Percentage  and  fractional  deviations  should  not  be  used 
when  applying  the  general  formulae  (10)  to  (12a),  (15) 
and  (16).  If  in  the  statement  of  a  problem,  as  in  example 
3,  the  fractional  or  percentage  precision  is  given,  the  cor- 
responding deviations  8  or  A  should  first  be  computed 
before  proceeding  with  the  solution. 

There  are,  however,  a  large  number  of  formulae  which 
may  be  discussed  with  a  great  saving  of  time  and  labor 
by  the  use  of  percentage  or  fractional  deviations.  This 
is  the  case  whenever  the  function  M  =/(rai,  m2,  .  .  .  mn) 
can  be  put  in  the  form  of  a  product  of  the  general  type 

M  =  k  .  mf  .m£  ...m*  (17) 

where  k,  a,  b,  :  .  .  p  are  constants  (positive,  negative, 
fractional,  or  integral).  For  all  such  cases  a  very  simple 
relation  holds  between  the  fractional  or  percentage  devia- 
tion in  any  component  and  the  fractional  or  percentage 
deviation  which  it  produces  in  the  final  result.  This 
may  be  shown  as  follows.  Applying  the  general  for- 
mula (10)  for  separate  effects  to  the  above  special  case, 
we  have  for  the  deviation  A!  in  M  produced  by  Si  in  mi 

9M 


Dividing  through  by  equation  (17) 

-£-0  .  ^  (18) 

M          mi 

CN 

i.e.,  a  fractional  deviation  --  in  mi  produces  a  fractional 


EQUAL   EFFECTS  35 

deviation  a  times  as  great  in  the  final  result.    Thus,  if 

8 

a  =  2,  a  deviation  of  one  per  cent.  (100  —  =  1)  in  mi  will 

tn\ 

introduce  a  deviation  of  two  per  cent,  in  M,  no  matter 
what  the  value  of  the  remaining  factors  in  the  expression 
may  be.  The  separate  effect  of  a  known  fractional  or 
percentage  deviation  in  any  component  on  the  final 
result  may  therefore  be  stated  at  once  by  inspection, 
whenever  the  formula  under  discussion  can  be  put  in 
the  above  form. 

Since  the  formula  for  Resultant  Effects  (12a)  may  be 
put  in  the  form 


M 

the  complete  solution  for  any  product  function  of  the 
type  given  by  equation  (17)  may  be  written  down  at 
once  by  inspection  as 


i 

if  the  fractional  deviations  — ,  etc.,  of  the  components 

?n\ 

are  known. 

Similarly,  the  solution  of  the  converse  problem  for  a 
product  function  is  equally  simple,  as  the  condition  for 
equal  effects,  page  32,  may  be  written 

_Al=^2_  _Ak_  __A^ 

M    M~          ~M~~  '••  ~M 

and  hence  the  allowable  fractional  deviation  in  the  final 
result  which  any  component  as  mk  may  produce  is 


M  <  sin   M 
where  TT  is  the  prescribed  fractional  deviation  of  the 


36  PRECISION   OF  MEASUREMENTS 

final  result  which  must  not  be  exceeded.     Having  thus 

determined  the  value  of  -A,  we  obtain  at  once  the  allow- 

g 
able  fractional  deviation  —  -  in  the  corresponding  com- 


ponent  mk  by  inspection  from  the  simple  relation  ex- 
pressed by  equation  (18). 

Since  the  exponents  a,  6,  c,  etc.,  of  the  factors  may 
have  negative  as  well  as  positive  values,  the  above  solu- 
tions apply  to  formulae  involving  division  as  well  as  mul- 
tiplication of  factors. 

It  is  to  be  especially  noted  that,  if  the  function  M  in- 
volves the  sum  or  difference  of  several  components,  or 
is  a  trigonometric  or  logarithmic  function,  no  simple 
relation  exists  between  the  fractional  deviation  of  a  com- 
ponent and  the  fractional  deviation  which  it  produces  in 
the  final  result.  The  above  special  method  of  procedure 
is,  therefore,  inapplicable  to  such  cases.  This  will  be 
readily  seen  from  the  following  simple  example.  Sup- 

pose 

M=  a  rrii+b  m2. 

Then  A' 

and  Ai 


from  which  it  appears  that  -^  stands  in  no  simple  relation 

<\ 
to  —  unless  the  term  b  m2  happens  to  be  negligible  in 

mi 

magnitude  compared  with  a  Wi,  in  which  case  we  should 
have  assumed  at  the  outset  for  our  precision  discussion 
that  M  =  ami  approximately. 

It  frequently  happens,  however,  that  apparently  com- 
plicated functions  can  be  transformed  into  a  simple 
product  of  factors  by  changing  variables  or  noting  that 
certain  components  may  be  neglected  in  the  precision 
discussion.  When  this  is  possible,  the  fractional  method 
may  be  applied  with  advantage  to  each  factor. 


EQUAL   EFFECTS  37 

Example  4. — The  solution  of  problem  2  may  be  obtained 
much  simpler  by  the  fractional  or  percentage  method  than  by 
the  general  method  as  worked  out  on  page  28.  For  the 

formula  g  =  7r2-  =  irH  t-3-  is  evidently  a  simple  product  func- 
tion of  the  variables  I  and  t.  To  find  the  deviation  in  g 
due  to  a  deviation  of  Si  =  0.10  cm.  in  I  and  a  deviation 
8t  =  0.0020  second  in  t,  we  find  first  the  fractional  deviation 
in  I  and  in  t  respectively. 

di      0.10  cm. 


/        100  cm. 


=  0.0010, 


and  ^  =  0.0020  sec.  = 

t          1.0  sec. 

Then  by  inspection,  since  g  is  directly  proportional  to  the 
first  power  of  Z, 

*'  =  5J  =  0.0010; 

and,  since  g  is  proportional  to  the  second  power  of  t  (neglect- 
ing sign), 

^  =  2-  =  2  X  0.0020  =  0.0040. 


=  V/(0.0010)2  +  (0.0040)2  =  ±0.0041 

or      A  =  ±0.0041  X  980  =2  =  ±4.0  -^ 
sec.  sec. 

which  is  practically  the  same  result  previously  obtained, 
the  slight  difference  arising  from  the  use  of  but  two  signifi- 
cant figures  in  the  computation. 

Example  5.—  Again,  the  solution  of  example  3,  page  33, 
may  be  simplified  by  using  the  fractional  method.  Thus,  if 
it  is  stipulated  that  g  is  to  be  measured  to  0.10  per  cent., 

i.e.,  100  —  ~0.10,  the  prescribed  fractional  deviation  is 
—  ^  0.0010.  Distributing  this  deviation  by  the  criterion  of 

equal  effects  between  the  component  measurements  I  and  t 
respectively,  we  have 


.  o.ooon. 

99^/29  y/2 

TT-l 

But  by  inspection  of  the  formula  g  =  /2  it  is  seen  that  g  is 


38  PRECISION    OF  MEASUREMENTS 

proportional  to  the  first  power  of  I  and  to  the  second  power 
of  t,  therefore 

A?      81        ,  At      0  8t 
—  =  v  and  —  =  2  —  • 
9       I  9          t 

Hence 

,-  =  0.00071, 

or  the  length  must  be  measured  to 

5z  =  100  cm.  x  0.00071  =  0.071  cm. 
Similarly, 

-'  =4  X  0.00071  =  0.00036, 

t          £i 

or  the  time  must  be  measured  to 

dt  =  1  sec.  X  0.00036  =  0.00036  sec. 

These,  it  is  seen,  are  practically  the  same  values  previously 
obtained  by  the  differentiation  method  on  page  33,  the  dif- 
ference in  the  second  place  of  figures  arising  from  the  use  of 
only  two  figures  in  the  computation. 

Discussion  of  "Equal  Effect"  Solution. — It  not  infre- 
quently happens  upon  solving  a  problem  as  described 
above,  that  some  component  (or  components)  can  with 
little  additional  time  and  labor  be  determined  with  a 
much  higher  precision  than  the  solution  demands.  In 
this  case  such  a  component  or  factor  should  be  so  meas- 
ured and  then  regarded  as  a  constant  in  the  precision 
discussion,  since  the  error  in  it  will  have  a  negligible 
effect  on  the  final  result.  The  problem  should  then  be 
re-solved  on  the  basis  of  one  less  variable,  in  which 
case  the  remaining  components  which  are  more  difficult 
to  determine,  may  be  measured  with  somewhat  less  pre- 
cision than  was  demanded  by  the  first  solution. 

The  proper  adjustment  of  precision  among  components 
so  as  to  give  the  desired  precision  in  the  final  result  with 
the  apparatus  at  one's  disposal  and  with  the  least  ex- 
penditure of  time  and  labor,  requires  some  experience 
and  good  judgment  on  the  part  of  the  investigator. 
Beginners  will  not  go  far  astray,  however,  if  they  follow 
the  above  criterion  for  equal  effects. 


PART  II. 
GRAPHICAL  METHODS. 


GRAPHICAL   METHODS. 


Nature  of  Problems — The  graphical  method  of  discussing 
experimental  data  is  of  great  convenience  and  importance 
when  the  problem  under  investigation  is  to  determine  the 
law  or  fundamental  relationship  between  two  quantities. 
This  type  of  problem  arises  very  frequently  in  scientific 
and  technical  investigations.  The  graphical  method  is 
also  of  great  value  for  purposes  of  interpolation,  discus- 
sion of  corrections,  etc. 

Procedure. — The  general  procedure  to  be  followed  in  dis- 
cussing observations  by  the  graphical  method  will  be  ex- 
plained and  illustrated  by  following  through,  step  by  step,  a 
specific  problem.  Suppose  it  is  desired  to  find  the  relation 
which  holds  between  the  resistance  of  a  certain  coil  of  wire 
and  its  temperature,  between  10°  and  100°  C.;  that  is,  to 
determine  the  formula  by  which  the  resistance  can  be  com- 
puted at  any  given  temperature  between  these  limits.  The 
experimental  procedure  would  consist  in  making  a  series 
of  measurements  of  the  resistance  r  of  the  wire  at  various 
temperatures  t  from  approximately  10°  to  100°  C.  Suppose 
that  the  result  of  such  experiments  gives  the  two  following 
columns  of  data,  the  resistance  measurements  being  reliable 
to  0.003  ohm,  and  the  temperature  measurements  to  0.02° 
C.,  as  shown  by  their  respective  deviation  or  precision 
measures. 

EXPERIMENTAL  DATA. 

r  =  resistance  of  t  =  temperature  of  coil 

coil  in  ohms.  in  degrees  C> 

10.421  10.50 

10.939  29.49 

11.321  42.70 

11.799  60.01 

12.242  75.51 

12.668  91.05 


42  GRAPHICAL    METHODS 

The  Direct  Plot. — To  obtain  some  clue  to  the  relation 
between  r  and  t  (supposing  it  unknown),  a  Direct  Plot 
should  first  be  made.  Plotting-paper  suitable  for  this 
work  should  be  ruled  with  carefully  adjusted  pens,  other- 
wise the  errors  arising  from*  irregularity  of  ruling  may 
easily  exceed  those  of  only  moderately  accurate  data.  A 
convenient  size  is  about  eight  by  ten  inches,  ruled  either 
in  millimeters,  or  preferably,  in  twentieths  of  an  inch. 

First. — Choice  of  Ordinates  and  A  bscissce.  The  first  thing 
to  decide  upon  is  which  data  are  to  be  plotted  as  ordinates 
and  which  as  abscissae.  The  usual  convention  of  analytic 
geometry  should  always  be  followed.  If,  as  in  the  problem 
under  consideration,  it  is  desired  to  obtain  a  relation  in 
which  r  is  expressed  as  a  function  of  t,  then  values  of  r 
should  be  plotted  as  ordinates  and  values  of  t  as  abscissae. 
If,  on  the  other  hand,  it  were  desired  to  obtain  a  formula 
for  computing  the  temperature  t  corresponding  to  any 
resistance  r,  as  in  resistance  pyrometry,  the  converse 
would  be  the  case. 

Second. — Choice  of  Scales.  By  the  "scale"  of  a  plot  is 
meant  the  ratio  of  the  number  of  units  (inches,  centi- 
meters, etc.)  of  the  plot  to  one  unit  of  the  data.  Scales 
of  both  ordinates  and  abscissa?  should  be  clearly  indicated 
on  the  plot.  Thus,  if  100°  is  plotted  so  as  to  extend 
over  10  inches,  the  scale  is  10" :  100°  =  1  : 10,  or  one-tenth. 
This  is  usually  expressed  as  1  inch  to  10  degrees.  In 
general,  it  is  not  feasible  to  choose  the  same  scale  for 
both  ordinates  and  abscissae,  nor  should  the  attempt  be 
made  to  have  the  origin  fall  on  the  plot.  If  equal  scales 
are  chosen  for  both  abscissae  and  ordinates,  the  locus 
of  the  data  is  likely  to  be  a  line  either  nearly  horizontal, 
in  which  case  the  precision  of  the  data  plotted  as  ordi- 
nates is  sacrificed,  or  nearly  vertical,  in  which  case  the 
same  is  true  of  the  abscissae.  Moreover,  the  intersection 
of  a  nearly  horizontal  line  with  lines  parallel  to  the  axis  of 
X  can  be  read  off  only  with  difficulty  and  liability  to  error, 
while  its  point  of  intersection  with  the  axis  of  Y  is  much 


DIRECT   PLOT  43 

more  definitely  defined.  In  order,  therefore,  to  preserve 
equal  precision  in  the  interpolation  of  both  co-ordinates,  the 
line  should  be  inclined  as  nearly  as  may  be  at  an  angle  of 
45°  with  both  axes.  Deviations  of  10°  or  so  to  either  side 
of  this  position  are  not  serious. 

The  scales  chosen  should,  furthermore,  be  convenient;  i.e., 
in  aiming  to  distribute  the  data  approximately  45°  across 
the  plotting-paper,  scales  of  one  inch  equal  to  1,  2,  4,  5,  or 
10  units  (or  these  units  multiplied  by  10 ±n  where  n  is  an 
integer),  should  be  chosen,  but  never  such  scales  as  one  inch 
to  3,  7,  6,  11  units.  The  latter  scales  make  plotting  not  only 
laborious,  but  very  liable  to  error,  whereas  the  former  scales 
permit  data  to  be  plotted  with  facility.  In  choosing  scales 
for  plotting,  the  student  should  guard  as  carefully  against 
adopting  excessively  large  scales  as  excessively  small  ones. 
In  the  latter  case  the  plot  will  be  cramped  and  the  precision 
of  the  data  sacrificed.  In  the  former  case  the  deviations 
of  the  data  from  the  general  law  which  they  follow  are  likely 
to  be  so  magnified  to  the  eye  that  it  is  difficult  or  impossible 
to  draw  a  representative  line.  Moreover,  such  plots  give  an 
exaggerated  idea  of  the  precision  of  the  data.  As  an  upper 
limit,  a  safe  rule  to  follow  is  to  adopt  a  scale  which  permits 
of  easy  interpolation  to  not  more  than  two  uncertain  places 
of  figures  in  the  data;  i.e.,  to  that  place  corresponding  to  the 
second  significant  figure  in  the  deviation  or  precision  meas- 
ure. This  rule  applies  particularly  to  data  extending  over 
narrow  numerical  limits,  to  corrections,  etc. 

In  the  present  problem  it  is  seen  that  the  extreme  variation 
of  r  is  about  2.3  ohms,  and  of  t,  90°.  The  scales  should  there- 
fore be  so  chosen  as  to  distribute  these  quantities  well  over 
the  paper.  Scales  of  1"  =  0.4  ohm  and  I"  =  10°  evidently 
fulfill  this  condition  and  are  at  the  same  time  convenient. 
It  is  evident,  however,  that  some  of  the  precision  of  the 
data  will  be  sacrificed  in  plotting  with  these  scales,  since  it 
is  impossible  on  a  plot  of  the  size  chosen  to  locate  the  last 
significant  figure  of  the  data  with  any  great  degree  of  pre- 
cision. It  should  also  be  noticed  that  the  origin  will  not  fall 


44  GRAPHICAL   METHODS 

on  the  plot.  This  is  not  at  all  necessary,  and  only  in  those 
cases  when  the  data  for  both  variables  simultaneously 
approach  small  values  (zero)  is  this  likely  to  be  the  case. 
It  is,  however,  desirable  (although  not  imperative),  that  the 
zero  value  of  the  abscissae  should  fall  on  the  plot,  in  order 
to  determine  the  intercept  of  the  curve  with  the  ordinate 
through  this  point  for  reasons  explained  below. 

Third. — To  plot  the  data.  Data  should  be  plotted  as  fol- 
lows. Locate  the  abscissa  of  the  first  point  along  the  axis 
of  abscissae  and  with  a  straight  edge  placed  vertically  through 
this  point  draw  a  fine  line  about  one-eighth  of  an  inch  long 
approximately  at  the  place  where  the  corresponding  ordinate 
is  to  be  located.  Then  locate  the  ordinate  along  the  axis 
of  ordinates,  and  draw  a  short  horizontal  line  intersecting 
the  first  line  drawn.  The  intersection  is  the  desired  position 
of  the  point.  Never  locate  the  data  by  dots,  as  the  preci- 
sion attainable  with  the  paper  is  not  only  sacrificed,  but  there 
is  also  much  greater  liability  to  error  in  the  operation  of 
plotting  itself  when  the  attempt  is  made  to  locate  both 
ordinate  and  abscissa  at  the  same  time. 

Fourth. — To  draw  the  "Best  Representative  Line."  The 
data  being  plotted,  the  next  step  is  to  draw  a  smooth  curve, 
the  equation  of  which  shall  best  represent  the  law  connecting 
the  two  variables  in  question.  Inspection  of  the  general 
form  of  this  curve  will  usually  give  valuable  information  as 
to  the  form  of  the  equation  sought. 

If  the  points  appear  to  lie  along  a  straight  line,  the  best 
representative  line  may  be  located  by  moving  a  stretched 
fine  black  thread  among  the  points  until  a  position  is  found 
such  that  the  points  lie  as  nearly  as  may  be  alternately  on 
either  side  of  the  thread  and  in  such  a  manner  that  the  points 
above  the  line  deviate  from  it  by  the  same  amount  as  those 
below.  The  exact  criterion  for  locating  the  best  represen- 
tative line  would  be  to  so  adjust  it  among  the  points  that  the 
sum  of  the  squares  of  the  deviations  of  the  points  above  the 
line  is  equal  to  the  sum  of  the  squares  of  the  deviations  of  the 
points  below  the  line.  (Criterion  of  Least  Squares.)  When 


DIRECT    PLOT 


45 


the  best  position  of  the  thread  has  been  found,  the  location 
of  two  points  through  which  it  passes  is  noted,  and  a  fine 
straight  line  is  then  ruled  through  these  points  with  a  hard 
pencil,  or,  better,  with  a  ruling  pen. 

If  the  points  cannot  be  uniformly  distributed  about  a 
straight  line,  but  deviate  systematically  from  it,  then  the 
best  representative  curved  line  is  to  be  drawn  with  a  French 
or  a  flexible  curve.  The  line  should  in  this  case  be  drawn 
as  before,  so  that  the  points  are  distributed  as  nearly  as  may 


pff 

^- 

it* 

1A 

>" 

xj 

12.4 

UL 

BE 

£J 

u 

UL 

£L 

R 

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A. 
p 

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IF 

±£ 

^ 

p 

^x 

+.01 
0 

1 

r- 

• 

p 

_ 

p 

^\ 

R 

_ 

nr 

P 

<^ 

/ 

u.2 

x 

x 

X 

x 

X 

/ 

P 

P^ 

ifr 

~ 

" 

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5 

~ 

x 

'• 

__ 

/" 

" 

tf 

~ 

-51 

•T 

-f 

xf 

t-- 

n 

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-J 

•^ 

X 

~~ 

-~" 

- 

*«» 

/ 

J 

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-*-- 

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Ti 

J^ 

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,0 

y          io"          eo'          Jo*         40*          so-          *o-          70*         60*         9o»        m 
Temperature. 
PLOT  1. 

be  on  alternate  sides.  From  the  form  of  the  resulting  curve 
its  equation  may  often  be  inferred.  The  next  step  is  to 
determine  the  equation  of  the  curve  by  transforming  it 
graphically  into  a  straight  line  by  some  one  of  the  special 
methods  of  transformation  described  below.  The  numeri- 
cal constants  in  the  equation  of  a  straight  line  can  always 
be  readily  determined  directly  from  the  plot. 

In  the  problem  under  consideration,  the  points  are  seen, 
Plot  I.,  to  lie  very  closely  along  a  straight  line  A'A",  the 


46  GRAPHICAL   METHODS 

deviations  from  the  line  being  of  an  irregular  and  not  of  a 
systematic  character.  The  relation  between  r  and  t  is  there- 
fore a  linear  one;  i.e.,  of  the  first  degree.  To  determine 
completely  the  function  r=f  (t),  we  have  to  find  the  numeri- 
cal value  of  the  constants  in  tjje  equation  of  this  line  A!A". 
Determination  of  the  Constants  of  a  Straight  Line. — The 
general  equation  of  a  straight  line  is 

y  =  ax  +  6,  (1) 

where  a  and  b  are  constants. 

The  constant  a  =  -^-  is  the  tangent  of  the  angle  0  which 
ax 

the  line  makes  with  the  axis  of  X.  The  value  of  a  cannot  in 
general  be  found  by  reading  off  the  angle  with  a  protractor 
and  looking  out  the  value  of  its  natural  tangent,  as  the  angle 
is  usually  distorted  owing  to  the  unequal  scales  used  in  plot- 
ting. To  determine  a,  read  off  the  value  of  the  ordinate 
and  abscissa  x',  y'  and  x",  y",  respectively,  of  any  two  points 
on  the  line,  preferably  near  the  extremities.  These  points 
will  not  in  general  be  observed  points.  Then 

y"  —  yf 
a  =  tan  0  ==  x»  _^' 

Thus  the  co-ordinates  of  two  such  points  A'  and  A"  are 
seen  to  be  x'  =  6.0°,  yf  =  10.30  ohms,  and  x"  =  94.5°, 
y"  =  12.76  ohms.  Hence 

12.76  —  10.30      2.46 


94.5   -    6.0       85 

The  constant  b  is  the  value  of  y  when  x  =  0;  that  is,  it  is 
the  intercept  of  the  line  (prolonged,  if  necessary)  on  the 
axis  of  Y,  read  off  on  the  scale  of  ordinates  chosen.  Thus 
from  the  plot  it  is  seen  that  the  line  A' A"  cuts  the  ordinate 
through  x  =  0°  at  b  =  10.13.  The  desired  equation  con- 
necting r  and  t  is  therefore 

r  =  0.0278  *  +  10.13.  (2) 


RECTIFICATION    OF   CURVES  47 

Whenever  the  data  are  such  that  a  long  extrapolation  of 
the  line  is  necessary  in  order  to  make  it  cut  the  ordinate 
through  x  =  0,  or  when  this  ordinate  falls  off  the  plot,  the 
value  of  b  is  found  as  follows.  Substitute  the  value  of  a 
as  determined  above,  together  with  values  of  x*  and  y'  of  some 
point  on  the  line,  in  equation  (1)  and  solve  for  b  directly. 

It  is  to  be  noted  here  that  the  precision  of  the  constants  in 
equation  (2)  is  less  than  the  precision  of  the  original  data. 
Values  of  r  computed  by  this  formula  cannot  at  best  be  more 
precise  than  one  or  two  parts  in  1,000,  while  the  observed 
values  were  stated  to  be  reliable  to  3  parts  in  10,000;  in 
other  words,  the  full  precision  of  the  data  has  not  been 
utilized  in  the  plot  of  the  size  here  chosen.  The  procedure 
by  means  of  which  the  precision  of  the  constants  as  above 
determined  may  be  increased,  and  another  place  of  signi- 
ficant figures  obtained,  will  be  explained  later.  See  Resid- 
ual Plot,  p.  60. 

Rectification  of  Curved  Lines. — When  the  plotted  data  do 
not  lie  along  a  straight  line,  the  form  of  the  smooth  curve 
best  representing  the  points  will  often  suggest  the  relation- 
ship sought.  Thus  curves  resembling  any  of  the  conic  sec- 
tions or  trigonometric  functions  are  usually  readily  recog- 
nized. In  all  such  cases  it  is  usually  necessary  to  transform 
the  curve  into  a  straight  line  in  order  to  determine  the  con- 
stants in  its  equation.  Suppose  from  inspection  of  the  curve 
that  the  relation  y  =  F  (x)  is  suggested.  If  F  (x)  can  be 
factored  or  written  in  the  form 

F  (x)  =  af(x)  +  b, 

where  a  and  b  are  numerical  constants  and  f(x)  contains  no 
constants,  the  function  suggested  can  be  very  readily  tested 
graphically.  This  includes  evidently  the  special  cases  when 
a  =  1  and  when  6  =  0;  i.e.,  the  functions, 

y  =  af(x);  y  =  f(x);  and  y  =  f(x)  +  b. 

In  all  of  these  cases  let  f(x)  =  z,  and  for  each  value  of  x  of 
the  data  compute  the  corresponding  value  of  z.  Construct 


48  GRAPHICAL  METHODS 

a  new  plot  with  values  of  z  as  abscissae  and  the  correspond- 
ing (observed)  values  of  y  as  ordinates.  The  general  equa- 
tion of  the  new  line  will  then  be 

y  =  az  +  b, 

jt 

or  that  corresponding  to  the  above  special  cases, 

y  =  az;  y  =  z;  and  y  =  z  +  &, 

all  of  which  are  equations  of  a  straight  line,  the  constants 
of  which  may  readily  be  determined  as  described  above. 
Whether  the  assumed  equation  y  =  af(x)  +  b  represents 
the  experimental  data  or  not  can  thus  be  judged  by  the  mag- 
nitude and  sign  of  the  deviations  of  the  plotted  data  from  the 
straight  line.  If  the  correct  function  has  been  assumed,  the 
values  of  the  constants  a  and  b  should  be  corrected  by  means 
of  a  residual  plot,  provided  the  precision  of  the  data  war- 
rants it. 

Problem. — Trigonometric  Functions.  Suppose  that  with  a 
certain  galvanometer  the  following  deflections  0  are  pro- 
duced by  the  currents,  7,  respectively,  and  it  is  desired  to  de- 
termine the  law  of  the  galvanometer,  i.e.,  the  form  of  the 
function  /  =  F  (6),  so  that  the  curre'nt  corresponding  to  any 
deflection  may  be  computed. 


OBSERVED   DATA. 

Deflection  6.  Current  I.  z  =  tan  6. 

10.17°  0.0704  0.1794 

19.27°  0.1368  0.3496 

29.16°  0.2184  0.5580 

40.47°  0.3348  0.8532 

48.45°  0.4430  1.128 

55.90°  0.5780  1.477 


The  data  plotted  directly  with  values  of  I  as  ordinates  and 
0  as  abscissas  are  found  to  lie  along  a  curve  A,  Plot  II.,  which 
evidently  suggests  the  relation  I  =  a  tan  0  where  a  is  con- 
stant ;  f or  /  =  0  when  6  =  0°  and  /  approaches  a  very  great 


TRIGONOMETRIC   FUNCTIONS 


49 


value  (infinity),  for  6  =  90°.  Comparing  the  suggested 
equation  with  y  =  af  (x),  we  see  /  (x)  =  tan  0.  Hence,  to 
test  the  suggested  equation,  we  compute  the  value  z  =  tan  0 
for  each  observed  value  of  0,  and  construct  a  new  plot  with 
the  values  of  /  as  ordinates  as  before,  and  values  of  z  as 


Deflections  6 — Curve  A. 
Tangent  0 — Curve  B 

PLOT  II. 


abscissae.  The  line  best  representing  these  data  is  shown 
in  B.  This  line  must  necessarily  pass  through  the  origin, 
since  the  current  and  corresponding  deflection  of  the  gal- 
vanometer approach  the  value  zero  simultaneously.  The 
galvanometer  is  seen  to  follow  the  law  of  tangents  between 
0°  and  60°.  Since  the  line  B  passes  through  the  origin,  the 
value  of  the  constant  a,  i.e.,  the  tangent  which  the  line 


50  GRAPHICAL  METHODS 

makes  with  the  axis  of  X,  is  readily  found  from  the  co-or- 
dinates x"  ?/"  of  a  single  point  M  to  be 


The  value  of  a,  in  this  particular  case,  can  also  be  obtained 
as  follows:  Since  tan  45°  =  1,  it  follows  from  y  =  atanfl 
that  y  =  a,  for  0  =  45°  or  z  =  1;  i.e.,  the  ordinate  of  curve 
A  at  45°  or  the  ordinate  of  curve  B  at  z  =  1  gives  the  value 
of  a  directly.  By  this  method  we  find  a  =  0.393  from  M', 
curve  A,  and  a  =  0.392  from  M",  curve  B,  both  values  being 
in  good  agreement  with  that  obtained  in  the  usual  way. 
The  desired  formula  for  the  galvanometer  is,  therefore, 

I  =  0.392  tan  0. 

If  the  observations  had  been  extended  beyond  0  =  60° 
and  it  were  found  that  the  points  corresponding  to  these 
data  regularly  deviated  from  the  straight  line,  the  conclusion 
would  be  that  the  instrument  followed  the  law  of  tangents 
only  within  certain  limits,  which  could  be  thus  determined. 

Problem.  —  Reciprocal  Functions.  Suppose  the  volume  v  of 
a  definite  mass  of  gas  kept  at  constant  temperature  is  de- 
termined at  various  pressures  p  with  the  following  results, 
and  it  is  desired  to  find  the  law  connecting  p  and  v;  e.g.,  to 
determine  the  form  of  the  function  p  =  /  (v). 

Pressure  p  Volume  v 

in  cm.  of  Hg.  in  c.  c.  v 

37.60  41.90  0.02380 

39.35  40.13  0.02493 

43.59  36.51  0.02739 

47.50  33.67  0.02971 

54.34  29.65  0.03373 

56.26  28.63  0.03497 

58.28  27.70  0.03610 

Constructing  a  direct  plot  from  these  data,  we  obtain  a 
slightly  curved  line  A,  Plot  III.  The  volume  diminishes  as 
the  pressure  increases,  but  not  proportionally,  since  the  data 
do  not  lie  along  a  straight  line.  The  curve  suggests  an  equi- 


RECIPROCAL   FUNCTIONS 


51 


lateral  hyperbola  referred  to  its  asymptotes  as  axes,  the 
equation  of  which  is  xy  =  const.  If  this  suggested  relation 
be  the  correct  one,  i.e.,  if  vp  =  const.,  or,  otherwise  written, 

p  =  const.  f-Y  by  changing  the  variable  from  v  to  z  =  -, 
the  resulting  equation  becomes  p  =  const,  z,  the  equation  of 


60 


SCALE 


DIRECT  PL|OT 
RECIPROCAL 


ORO. 
ABS. 
PLOT    ORD. 


38 


I  •  4  CM. 
I"-  2CC. 
l"=  -4  CM. 
\"m  0.002 


.020 


.030 


.09* 


Curve  A, — Volumes. 

Curve  B,— Reciprocal  of  Volumes 

PLOT   III. 


a  straight  line.  Constructing,  therefore,  a  second  plot,  with 
the  same  values  of  p  as  ordinates  and  the  reciprocal  values 
of  v  as  abscissae,  we  obtain  curve  B,  which  should  be  a  straight 
line  if  the  gas  in  question  follows  Boyle's  law  within  the 
errors  of  the  experiment.  This  is  seen  to  be  the  case.  If 
it  were  not  the  case,  a  study  of  the  deviations  of  the  data 


52  GRAPHICAL   METHODS 

from  the  straight  line  would  afford  a  proper  means  of  dis- 
cussion of  the  deviations  of  the  gas  from  Boyle's  law. 

Another  method  of  treating  this  problem  would  be  to 
compute  the  product  pv  for  each  pair  of  values  of  p  and  v 
and  then  to  discuss  .the  values  of  the  product  graphically 
as  follows.  With  values  of  p  as  abscissae  construct  a  plot 
with  corresponding  values  of  pv  as  ordinates.  If  the  data 
satisfy  •  the  relation  pv  =  const,  within  the  experimental 
error,  the  best  representative  line  will  be  parallel  to  the 
axis  of  abscissae,  with  the  values  of  pv  distributed  alternately 
and  about  equally  on  either  side.  If,  on  the  other  hand, 
the  gas  deviates  from  Boyle's  law,  as  many  gases  do  even 
under  ordinary  conditions,  and  as  all  gases  do  at  very  great 
values  of  p,  the  resulting  curve  will  give  information  not 
only  as  to  the  amount  of  the  deviations,  but  also  as  to  the 
method  of  correcting  the  assumed  simple  relationship  to 
make  it  better  conform  with  experimental  facts. 

The  Logarithmic  Method.  Exponential  Functions. — If  the 
data  of  a  direct  plot  are  found  to  deviate  continually  from 
a  straight  line,  they  may  very  often  be  represented  by  an 
exponential  equation  of  the  form  y  =  mxn  where  m  and  n 
are  constants  which  may  have  any  value.  Cases  of  this 
kind  are  of  very  frequent  occurrence,  and  it  is,  therefore, 
of  great  importance  to  be  able  to  test  this  relationship  and 
to  determine  the  numerical  values  of  the  constants  m  and 
n.  This  can  always  be  done  by  means  of  a  Logarithmic 
Plot;  i.e.,  a  plot  constructed  with  the  values  of  the  loga- 
rithms of  y  as  ordinates  and  the  corresponding  logarithms 
of  x  as  abscissae.  For,  if  we  take  logarithms  of  both  sides 
of  the  equation  y  =  mx",  we  have 

log  y  =  n  log  x  -\-  log  m. 

Changing  the  independent  variables  x  and  y  in  this  equation 
to  x'  and  y'  respectively,  by  putting  x'  =  log  x  and  y'  =  log  y, 
and  writing  b  =  log  m,  the  equation  becomes  y'  =  nxf  -|-  &. 
This  is  the  equation  of  a  straight  line  of  which  the  in_ 
tercept  on  the  axis  of  Y  is  b  =  log  m,  and  of  which  the 


LOGARITHMIC   METHOD  53 

natural  tangent  of  the  undistorted  angle  which  it  makes  with 
the  axis  of  X  is  n.  Hence  the  constants  in  the  -original  equa- 
tion y  =  mxn  may  be  obtained  at  once  by  looking  out  the 
number  ra  whose  logarithm,  b,  is  the  intercept  of  the  straight 
line  on  the  axis  of  Y,  and  by  determining  the  tangent  which 
the  line  makes  with  the  axis  of  X. 


93 


ECT  PLOT 
6.  PLOT 


GALE 
ORD.  l" 
ABS.  I" 
ORD.  I" 
ABS  ! 


VVO  CM. 
O.ISECONJD 
O.2000 
O.I  000 


J20 


. 

T* 


T6 

PLOT   IV. 


T.7 


0.6 
10 


O.7 
Tft 


0* 


Here,  again,  the  values  of  the  constants  m  and  n  as  thus 
determined  are  usually  reliable  to  not  more  than  0.5%,  and 
hence,  if  the  orginal  data  warrant  it,  they  should  be  further 
corrected  by  means  of  a  residual  plot. 

Problem. — The  logarithmic  method  will  now  be  illustrated 
by  discussing  data  obtained  for  a  body  falling  freely  under 
the  influence  of  gravity.  Suppose  experiments  gave  the  fol- 
lowing values  for  the  distance  s,  through  which  a  ball  fell 
in  the  time  t,  and  it  is  desired  to  deduce  the  law  between  s 


54  GRAPHICAL   METHODS 

and  t;  i.e.,  to  find  the  equation  by  which  the  distance  s  can 
be  computed  for  any  value  of  t. 

OBSERVED  DATA. 

Distance  s  in  Time  tin  sf  =  log  s.  t'  =  log  t. 

centimeters.  seconds.         ^ 

30.13  0.2477  1.4790  1.3939 

85.26  0.4175  1.9308  I-6207 

150.39  0.5533  2.1772  L7430 

223.60  0.6760  2.3495  1.8300 

274.20  0.7477  2.4381  1.8737 

A  direct  plot  A,  Plot  IV.,  of  s  and  t,  shows  at  once  that  s 
and  t  are  not  proportional.  The  regular  deviation  from  a 
straight  line  suggests  an  exponential  curve,  i.e.,  s  =  ml". 
To  test  this  relation,  we  construct  on  ordinary  co-ordinate 
paper  a  "logarithmic  plot"  B  with  s'  =  log  s  as  ordinates 
and  if  =  log  t  as  abscissae.  Convenient  scales  which  dis- 
tribute these  values  about  45°  across  the  paper  are  V  =  0.2 
for  the  ordinates  and  1"  =  0.1  for  the  abscissae.  It  is  to  be 
noticed  that  the  values  of  t,  being  less  than  unity,  lead  to 
values  of  log  t  with  negative  characteristics.  The  abscissae 
are,  therefore,  laid  off  to  the  left  of  the  origin  as  indicated, 
the  plot  thereby  lying  in  the  second  quadrant.  The  data 
are  seen  to  lie  very  closely  along  a  straight  line,  the  constants 
of  which  are  to  be  determined  as  described  on  page  46.  Thus 
the  intercept  of  the  line  on  the  axis  of  Y  is  b  =  log  m  =  2.688, 
whence  m  =  488.  The  tangent  which  the  line  makes  with 
the  axis  of  X  is  found  to  be  n  =  1.995  or  n  =  2.00  within 
the  error  of  plotting. 

The  desired  equation  is,  therefore,  s  =  488  t2-00.  Since  the 
law  of  falling  bodies  is  known  to  be  s  =  Jgtf2,  it  follows  that 
%g  =  488,  or  the  mean  value  of  g  from  the  data,  within  the 

cm 
error  of  direct  plotting,  is  g  =  2  X  488  =  976  —— ^ . 

sec. 

If  the  data  s  and  t  are  reliable  to  more  than  about  0.5  per 
cent.,  the  constants  m  and  n  should  be  corrected  by  means 
of  a  residual  plot. 

Attention  should  be  called  to  one  important  point  in  this 


LOGARITHMIC   METHOD  55 

connection.  In  constructing  a  plot  like  the  above  in  which 
the  intercept  on  the  axis  Y  is  to  be  determined,  it  is  con- 
venient to  choose  the  units  in  which  the  abscissae  are  ex- 
pressed such  that  the  resulting  line  cuts  the  axis  of  Y  without 
a  long  extrapolation.  By  a  suitable  choice  of  units  this 
condition  can  always  be  attained,  for  increasing  or  dimin- 
ishing the  unit  expressing  the  abscissas  by  a  multiple  of  ten 
does  not  affect  the  slope  of  the  line,  but  simply  shifts  it 
parallel  with  itself  to  or  from  the  origin. 

Logarithmic  Plotting-paper. — When  the  constants  of  a  num- 
ber of  exponential  curves  of  the  type  y  =  mxn  are  to  be 
determined,  a  great  saving  of  time  and  labor  may  be  effected 
by  using  so-called  logarithmic  co-ordinate  paper.  Four 
quadrants  of  such  paper  are  shown  in  Plot  V.  The  length 
OX  is  laid  off  equal  to  OF  and  put  equal  to  10  or  some  in- 
tegral power  of  10  units.  This  is  then  subdivided  into  spaces 
such  that  the  distances  1-2,  1-3,  1-4,  etc.,  are  proportional 
to  the  logarithms  of  2,  3,  4,  etc.  Thus  the  point  numbered 
2  is  located  not  at  two-tenths  the  distance  from  0  to  Z, 
as  in  ordinary  plotting-paper,  but  at  log  2  =  0.301  of  the 
distance  OX.  The  rulings  thus  become  more  and  more 
crowded  together  as  they  proceed  from  0  to  X  and  Y.  Con- 
tinuing the  rulings  beyond  10  in  either  direction,  it  is  evi- 
dent that  the  unit  square  XOY  repeats  itself  indefinitely, 
since  the  value  of  the  logarithm  of  any  quantity  multiplied 
by  10*,  where  &  is  a  positive  or  negative  integer,  is  equal 
to  the  logarithm  of  the  original  quantity  plus  k.  Thus 
the  point  marked  0.2  is  laid  off  at  a  distance  equal  to 
log  0.2  =  log  (2  X  10-1)  =  log  2  --  1  =  —  1  +  0.301  to 
the  left  of  0;  i.e.,  just  the  distance  OX  to  the  left  of  the 
point  marked  2,  etc. 

It  is  evident  that  a  series  of  values  of  x  and  y  which  satisfy 
the  equation 

y  =  mx* 

will,  if  plotted  directly  on  logarithmic  paper,  lie  along  a 
straight  line;  for  the  paper  has  the  effect  of  locating  the 
data  y  and  x  at  points  proportional  to  their  logarithms,  and 


56 


GRAPHICAL   METHODS 


it  has  been  shown  on  page  52  that  this  leads  to  a  straight 
line.  We  are  thus  saved  the  labor  of  looking  out  logarithms 
and  locating  them  on  rectangular  co-ordinate  paper  as  pre- 
viously explained.  Moreover,  since  the  scales  of  ordinates 
and  abscissae  are  here  necessarily  equal,  the  slope  of  the 


f 

9 

ft 

7 

6 

5 

/ 

4- 

/ 

3 

/ 

/ 

2 

4 

&      . 

J 

4/ 

/ 

5 

1 
<§ 

.7 

s 

.? 

f 

2 

3 

4 

5 

7 

8  ,Q 

V' 

/ 

.9 

/ 

•ft- 

1 

g 

/ 

.7 

/ 

.6 

/ 

£ 

> 

' 

,4         ORD: 

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ET    f 

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re 

R  >. 

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A5S: 
3 

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s. 

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Y' 

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PLOT  V. 

resulting  line  is  undistorted,  and,  therefore,  the  tangent 
which  it  makes  with  the  axis  of  X  is  obtained  by  measur- 
ing off  with  a  scale  the  distance  y" —  y'  and  x"  —  x'  of  two 

y"  y' 

points  on  the  line  and  taking  their  ratio,    ,, ,  =  n. 

gives  the  desired  value  of  the  exponent  of  x. 


This 


LOGARITHMIC   METHOD  57 

To  find  the  value  of  the  constant  m,  we  note  that  in  the 
equation 

log  y  =  n  log  x  -f-  log  m 

when  x  =  1, 

log  y  =  log  m. 

Hence,  since  values  read  off  on  logarithmic  paper  correspond 
to  the  numbers  of  which  the  spacings  are  the  logarithms, 
the  intersection  of  the  logarithmic  plot  with  the  ordinate 
through  x  =  1  gives  at  once  the  value  of  m,  whereas  with 
rectangular  plotting  paper  the  intercept  with  the  Y-axis 
gives  the  value  of  log  m. 

The  following  difference  between  logarithmic  plots,  drawn 
on  rectangular  and  on  logarithmic  paper,  should  also  be  noted. 
Suppose  the  line  representing  the  data  plotted  on  logarith- 
mic paper  intersects  the  ordinate  not  through  x  =  1,  but 
through  x  =  10*,  where  k  is  any  positive  or  negative  in- 
teger. Let  ra'  be  the  value  of  the  intercept.  To  obtain 
the  value  of  m  in  the  equation  y  =  mxn  from  m',  we  have 
y  =i  m'  for  x  =  10*,  i.e., 

m'  =  m  10*" 

mf 

= 


In  the  case  of  rectangular  paper,  on  the  other  hand,  the 
line  intersects  the  ordinate  through  x  =  log  10*  =  k  at  m". 
To  obtain  m  from  this  intercept,  we  have 

w"  =  log  y  =  nk  log  10  +  log  m 

=  nk  -f-  log  m 
Hence  log  m  =  m"  —  nk. 

The  solution  of  the  problem  discussed  on  page  54,  by  the 
use  of  logarithmic  paper  is  shown  on  a  reduced  scale  in  Plot 
V.  Values  of  s  and  t  are  plotted  directly  and  lie,  as  is  seen, 
along  a  straight  line.  It  is  convenient  to  express  here  values 
of  s  in  meters  instead  of  centimeters.  The  tangent  which 
this  line  makes  with  the  axis  of  X  measured  off  directly 
along  OY  and  OX'  is  found  to  be  2.00.  Extrapolating  the 


58  GRAPHICAL   METHODS 

line  to  cut  the  ordinate  OY  through  x  =  1,  we  see  the  inter- 
cept to  be 

m  =  4.89. 

The  equation  connecting  s  and  t  is  therefore 

s=,4.89Z2-°° 

which  agrees,  within  the  error  of  plotting,  with  that  pre- 
viously obtained  with  rectangular  co-ordinate  paper,  when  we 
remember  that  in  the  above  equation  s  is  expressed  in  meters 
instead  of  centimeters. 

Equations  of  the  Form  y  =  m  (x  -f-  ft)". 

A  slightly  more  complicated  relation  than  that  represented 
by  the  equation  y  =  mxn,  which  may  also  be  treated  by  the 
logarithmic  method,  is  that  represented  by  the  formula 

y  =  m  (x  +ft  Y 

where  ft  is  a  constant.  If  a  logarithmic  plot  be  made  with 
data  xfli,  z22/2>  etc.,  which  satisfy  an  equation  of  this  form, 
the  points  will  not  lie  along  a  straight  line  for  both  large  as 
well  as  small  values  of  the  variables.  Suppose  it  is  found 
that  for  large  values  of  x  and  y  the  curve  is  practically 
straight,  but  for  small  values  it  becomes  curved.  Under 
these  circumstances  it  is  worth  while  to  see  if  the  best  repre- 
sentative logarithmic  plot  cannot  be  rectified  into  a  straight 
line  by  assuming  an  equation  of  the  above  form,  for  by 
putting  z  =  x  +  ft  the  equation  reduces  to 

y  =  mzn 

from  which  m  and  n  are  easily  determined,  if  values  of  y 
and  z  are  plotted  logarithmically.  It  is  only  necessary 
therefore  to  find  the  value  of  the  constant  ft  to  be  added  to 
all  values  of  x.  This  may  be  found  as  follows.  Select  two 
points  a;^  and  x%y2  near  the  ends  of  the  original  logarithmic 
plot.  Compute  the  ordinate  2/3  of  an  intermediate  point 
such  that 

log  2/3  =  i -Logj/1  +  i  log  2/2 
or  2/3  =  V  2/i2/2 
and  look  out  its  corresponding  abscissa  XB  on  the  line. 


PRECISION   OF  PLOTTING  59 

Then  it  follows,  if  the  data  satisfy  an  equation  of  the  form 
=  m  (x  +  £)*,  that 

log  fa  +  ft  =  *  log  fa  +  £)  +  *  kg  fa  +  ft 
or  *3  +  j8  =  V 

2 

from  which     = 


Having  thus  obtained  £,  proceed  in  the  usual  manner  to 
determine  m  and  n  from  a  new  logarithmic  plot  of  the  equa- 
tion y  =  m2n  where  2  =  x  +  ft. 

Equations  of  the  Form  y  =  m  10"*;  y  =  m  c** 
Data  satisfying  equations  of  the  form 

-^  =  m  I0nx  and  y  =  m  e**, 

where  e  is  the  base  of  Naperian  logarithms,  may  also  be 
treated  graphically  by  the  following  special  logarithmic 
method.  Taking  logarithms  of  these  equations,  we  obtain 

log  y  =  nx  -{•  log  m 
and         log  y  =  Mnx  -f-  log  m 

respectively,  where  M  =  0.4343  is  the  modulus  for  reducing 
Naperian  to  common  logarithms.  If,  on  ordinary  plotting 
paper,  values  of  y'  —  log  y  are  plotted  as  ordinates  and  the 
unchanged  values  of  x  as  abscissae,  the  resulting  curves  will 
be  straight  lines;  the  intercept  on  the  axis  of  Y  f  or  x  =  0 
will  give  yf  =  log  m  and  the  tangents  of  the  lines  with  the 
axis  of  X  will  give  the  values  of  n  and  Mn  respectively. 

Precision  of  Plotting.—  The  question  now  arises  as  to  the 
precision  of  the  constants  deduced  from  a  direct  or  rectified 
plot.  In  discussing  this  question,  we  will  consider  only  errors 
inherent  in  the  process  of  plotting  and  interpolation,  and  in 
the  plotting-paper  itself. 

The  error  of  estimating  tenths  of  the  smallest  division, 
together  with  the  uncertainty  introduced  by  the  width  of 
the  lines  locating  the  data  and  the  inaccuracies  in  the  paper 
due  to  errors  in  ruling  and  unequal  shrinkage,  make  0.02 
inch  a  fair  estimate  of  the  extreme  precision  of  reading  or 


60  GRAPHICAL   METHODS 

plotting.  If  the  plot  be  10  inches  on  a  side  (about  the  maxi- 
mum size  ordinarily  employed),  the  fractional  precision 

attainable  cannot   therefore  be  greater  than  about  -~  = 

0.002,  or  0.2  per  cent.  A  more  probable  estimate  of  the 
precision  ordinarily  attained  in  direct  plots  is  0.4  to  0.5  per 
cent.  Constants  deduced  from  a  direct  plot  of  the  size 
considered  can  therefore  be  relied  upon  only  to  this  degree 
of  precision;  that  is,  in  general,  to  three  significant  figures, 
with  the  fourth  doubtful. 

If  the  experimental  data  are  reliable  to  four  or  more  signifi- 
cant figures  (i.e.,  to  0.1  per  cent,  or  better),  some  of  the  pre- 
cision will  evidently  be  sacrificed  in  the  direct  plot  unless  a 
much  larger  plot  be  made.  In  order  that  the  full  precision 
of  such  data  may  be  utilized,  the  direct  plot  should  be  fol- 
lowed by  a  so-called  residual  plot,  by  means  of  which  the 
constants  first  obtained  can  be  corrected  and  rendered  more 
precise.  By  this  procedure  the  precision  of  the  graphical 
method  may  be  greatly  extended.  The  procedure  to  be  fol- 
lowed in  constructing  a  residual  plot  will  now  be  considered. 

Residual  Plot. — A  residual  plot  is  one  in  which  the  de- 
viations of  the  observed  data  from  the  "best  representa- 
tive line"  are  plotted  on  an  enlarged  scale.  It  serves  to 
correct  the  position  of  this  line  among  the  points,  to  correct 
the  numerical  value  of  the  constants,  and  to  test  whether 
the  data  follows  the  assumed  law  within  the  precision  of  the 
measurements.  It  is  constructed  as  follows.  Substitute 
in  the  equation  y  =  ax  +  6,  deduced  for  the  best  represen- 
tative line,  which  may  be  either  a  direct  or  rectified  plot, 
the  observed  values  of.  x,  and  compute  the  corresponding 
values  of  y.  The  differences  between  these  computed  values 
of  y  and  the  corresponding  observed  values  are  called  the 
residuals.  A  study  of  the  sign  and  magnitude  of  these  re- 
siduals furnishes  much  valuable  information  regarding  the 
representative  character  of  the  "best  line"  chosen,  and  the 
graphical  discussion  of  these  constitutes  the  residual  plot.  If 
a  plot  be  made  (preferably  on  the  same  paper  and  with  the 


RESIDUAL   PLOT  61 

same  scale  of  abscissae  as  the  straight  line  plot),  with  the 
values  of  the  residuals  r  =  y  (observed)  —  y  (computed)  as 
ordinates,  and  the  corresponding  values  of  x  as  abscissae,  we 
obtain  a  graphical  representation  of  the  deviations  of  the 
observed  data  from  the  line  assumed  to  best  represent  them. 
To  better  study  these  deviations,  they  should  be  plotted  on 
a  large  scale.  In  effect,  the  process  is  to  project  the  "best 
representative  line"  horizontally  and  to  magnify  the  devia- 
tions of  the  plotted  data  from  it.  If  it  is  found  that  the 
plotted  residuals  lie  alternately  and  about  equal  distances  on 
either  side  of  the  horizontal  line  passing  through  the  zero  of 
the  residuals,  the  conclusion  is  that  the  original  line  is  the 
best  line  which  can  be  drawn  to  represent  the  data.  In  gen- 
eral, however,  it  will  be  found  that  a  new  line  can  be  drawn 
among  the  residuals  which  will  distribute  them  more  nearly 
alternately  on  either  side.  The  values  of  the  tangent  a  and 
intercept  6,  found  for  the  original  representative  line,  should 
therefore  be  corrected  by  the  values  of  the  tangent  and  in- 
tercept respectively  of  the  new  best  representative  line  of  the 
residual  plot,  read  off  of  course  on  the  scales  on  which  it  is 
plotted.  In  this  way  the  original  constants  may  be  corrected 
to  the  fourth  significant  figure.  It  is  sometimes  necessary 
to  follow  the  first  by  a  second  residual  plot  when  extreme 
precision  is  desired. 

If  the  residuals  are  found  to  deviate  systematically  from 
the  straight  line,  the  conclusion  is  that  the  data  cannot  be 
represented  by  the  line  in  question  within  the  precision  of 
the  measurements.  In  such  a  case  a  new  formula  should  be 
sought. 

Illustration  of  a  Residual  Plot.  The  procedure  to  be  fol- 
lowed in  making  a  residual  curve  or  plot  will  be  illustrated 
by  the  data  given  in  the  Problem  discussed  in  Plot  I.,  p.  45. 
The  equation  of  the  best  representative  straight  line  for  these 
data  was  found  to  be 

r  =  0.0278  t  +  10.13. 

To  test  whether  this  equation  is  the  best  which  can  be 
obtained  to  represent  the  given  data,  we  proceed  to  compute 


62 


GRAPHICAL   METHODS 


the  residuals,  as  described  above,  by  substituting  the  observed 
values  of  t  and  computing  /. 


t  observed. 

r  observed. 

T*  computed. 

r-r1  first 
residuals. 

r*  computed. 

r-r"  second 
residuals. 

10.50 

10.421 

10.422 

m 

—0.001 

10.413 

+  0.008 

29.49 

10.939 

10.949 

—0.010 

10.946 

—0.007 

42.70 

11.321 

11.317 

+0.004 

11.317 

+0.004 

60.01 

11.799 

11.798 

+0.001 

11.802 

—0.003 

75.51 

12.242 

12.229 

+0.013 

12.237 

+0.005 

91.05 

12.668 

12.661 

+0.007 

12.673 

—0.005 

2  (r— r')2  =  437 


2  (r— r")2  =  188 


Inspection  of  these  residuals,  column  4,  affords  valuable 
information,  but  they  can  be  better  studied  graphically, 
especially  if  the  number  of  observations  is  great.  The  scale 
to  be  chosen  for  the  ordinates  should  not  be  greater  than 
about  1  inch  to  0.01  ohm,  since  this  will  permit  the  residuals 
to  be  plotted  directly  without  interpolation  to  the  last  place 
of  significant  figures  of  the  data,  while  by  estimation  the  plot 
can  be  read  to  the  next  place  of  figures;  i.e.,  to  0.0001  ohm, 
which  is  more  than  ten  times  the  precision  of  the  data.  The 
plot  may  conveniently  be  made  on  the  same  sheet  as  the  direct 
plot,  using  the  same  scale  of  abscissae  as  shown  in  Plot  I., 
p.  45.  The  heavy  horizontal  line  through  0  represents  the 
line  A' A!'  projected  horizontally.  The  residuals,  plotted  on 
a  magnified  scale,  are  connected  by  dotted  lines.  Inspection 
shows  that  the  positive  residuals  preponderate,  and  that  a 
new  line  B'B"  can  be  drawn  which  will  distribute  the  residuals 
more  nearly  alternately  on  either  side  of  it.  The  original 
line  A' A!'  should  evidently  have  been  drawn  with  a  slightly 
greater  inclination.  The  value  of  the  intercept  of  the  new 
line  B'B"  on  the  axis  of  Y  (on  the  scale  of  residuals)  is — 0.011. 
The  tangent  of  the  angle  which  it  makes  with  the  axis  of  X 
is  obtained  from  the  ordinates  and  abscissae  of  two  points 
B'  and  B"  on  the  line  respectively:  thus  xf  —  5.00°,  y'  = 
0.0095;  and  x"  =  95.05°,  y"  =  0.0135.  Hence 
y"-y'  _.  0.0135 -(-0.0095)  =_ 
F^"?  ~  95.05-5.00 


INTERPOLATION   FORMULA  63 

Hence   the  constants    of  the  original  equation  should  be 

corrected  by  these  amounts,  thus  becoming 
V  =  10.13  —  0.011  =  10.119 
a!  =  0.0278  +  0.00025  =  0.02805. 

The  corrected  equation  connecting  r  and  t  is,  therefore, 
r  =  0.02805  t  +  10.119. 

This  represents  the  original  data  much  better  than  the  first 
equation  obtained,  as  may  be  seen  from  the  sign  and  magni- 
tude of  the  new  set  of  residuals  r  —  r"  computed  from  the 
corrected  equation  and  given  in  the  last  column  of  the  table. 
There  is  now  seen  to  be  no  systematic  deviation  among  the 
residuals,  and  the  sum  of  their  squares  is  seen  to  be  much  less 
than  in  the  case  of  the  residuals  from  the  first  equation. 

Interpolation  Formulae.— It  frequently  happens  that  ex- 
perimental data  whose  locus  differs  slightly  but  progres- 
sively from  a  straight  line  cannot  be  represented  by  a  two 
constant  formula  of  the  general  exponential  form  y  —  mxn. 
This  is  the  case,  for  example,  with  data  on  the  coefficient 
of  expansion  of  many  substances  over  wide  ranges  of  tem- 
perature. To  obtain  an  algebraic  relation  for  such  cases, 
interpolation  formulae  of  the  general  form 

y  =  a  +  bx  +  ex2  +  dx3  +  .  .  . 

are  usually  assumed.  The  number  of  terms  to  be  taken  in 
this  equation  (i.e.,  the  number  of  constants  to  be  determined) 
depends  upon  the  precision  of  the  data  and  on  the  extent  of 
the  deviation  of  the  curve  representing  them  from  a  straight 
line. 

The  values  of  the  constants  in  such  an  equation  are  in  gen- 
eral best  determined  analytically.  For  this  purpose  it  is 
necessary  to  know  at  least  as  many  pairs  of  values  of  x  and 
y  as  there  are  constants  to  be  determined.  Thus,  if  the 
equation  assumed  to  represent  the  data  be  y  =  a  -f-  bx  +  ex2, 
it  is  necessary  to  know  at  least  three  pairs  of  values  of  x  and 
y  which,  substituted  in  the  equation,  will  lead  to  three 
simultaneous  equations,  from  which  the  values  of  the  three 
unknown  constants,  a,  &,  c,  can  be  at  once  determined  by 


64  GRAPHICAL   METHODS 

elimination.  It  is  seldom  necessary  to  carry  the  series  be- 
yond the  fourth  term  da?;  in  fact,  three  terms  are  sufficient 
for  most  purposes. 

In  general,  however,  the  experimental  data  furnish  many 
more  pairs  of  values  of  x  and  y  than  there  are  constants  to 
be  determined.  In  all  such  cases  the  most  probable  value  of 
the  constants  can  be  determined  by  the  graphical  procedure 
described  below  or  by  the  method  of  Least  Squares. 

Graphical  Solution.— Let    x^l}    x^,    .  .  .  xnyn,    be    the 
numerical  values  of  pairs  of  observations  on  the  variables  x 
and  ?/,  which  are  assumed  to  satisfy  the  equation 
y  —  a  +  bx  +  ex2. 

Any  three  pairs  of  values  substituted  in  this  equation  will 
give  three  simultaneous  equations  from  which  a,  b,  and  c 
can  be  computed,  but  the  values  of  these  constants  will  vary 
to  a  certain  extent  according  to  which  sets  of  values  of  x  and 
y  are  chosen.  The  simplest  procedure  by  which  to  obtain 
the  best  or  most  probable  values  of  a,  6,  and  c,  is  to  plot  all 
values  of  x  and  y  and  draw  the  best  representative  line 
among  them.  Then  select  three  points  on  this  line, — one 
near  each  end  and  one  half-way  between  for  convenience, — 
determine  their  ordinates  and  abscissae,  and  with  these  three 
pairs  of  values  form  three  simultaneous  equations  and  com- 
pute a,  b,  and  c.  Having  obtained  the  constants  in  this 
manner,  they  may  be  further  corrected  by  computing  re- 
siduals and  studying  these  by  means  of  a  residual  plot, 
although  this  requires  both  care  and  judgment.  A  more 
exact  although  more  laborious  method  of  procedure  is  the 
analytical  solution  of  the  equation  by  the  method  of  Least 
Squares. 

Least  Square  Solution — As    before,  let  Xjylt  z27/2,  •  •  • 
xnyn,  be  numerical  values  of  the  observations,  and 
y  =  a  -f-  bx  +  ex2 

the  equation  the  constants  of  which  are  to  be  determined. 
This  may  be  written 

y  —  a  —  bx  —  ex2  =  0. 


LEAST   SQUARE    SOLUTION  65 

If  the  observed  values  x,  y,  were  free  from  all  experimental 
errors  and  the  equation  represented  the  law  connecting  them, 
each  pair  would  exactly  satisfy  the  equation,  with  proper 
numerical  values  of  the  constants.  This,  however,  is  not  the 
case,  since  all  observations  are  liable  to  indeterminate  error. 
Hence,  if  the  observations  be  substituted  in  the  equation,  the 
right  member  will  not  in  general  equal  zero,  but  will  differ 
from  zero  by  some  small  quantity  v  called  the  residual  error, 
which  may  be  plus  or  minus.  Thus,  by  substituting  the 
observations  in  the  assumed  equation,  we  get  the  following 
so-called  "observation  equations": — 

yl  —  a  —  bxl  —  cxf  =  vlt 
—  a  —  bx  —  cx2  =  v 


yn a bXn CXn*  =  V», 

from  which  the  most  probable  values  of  the  constants  a,  6, 
c,  are  to  be  determined.  By  the  principle  of  Least  Squares 
those  values  of  a,  6,  c,  are  the  most  probable  which  make 
the  sum  of  the  squares  of  the  residual  errors  v  a  mini- 
mum; i.e.,  those  which  make  the  value  of  2,v2  =  v-f  + 
i>22  +  .  .  .  Vn2  a  minimum.  The  expression  2,v2  is  a  function 
of  the  quantities  a,  6,  c,  and  the  condition  that  it  shall  be  a 
minimum  is  that  its  first  differential  coefficient  with  respect 
to  these  variables  shall  be  zero,  and  its  second  differential 
coefficient  positive.  The  latter  test  need  not  be  applied, 
however,  as  inspection  will  distinguish  between  maximum 
and  minimum  values,  the  limit  of  the  former  being  evidently 
infinity. 

Applying  this  condition  to  the  above  observation  equa- 
tions, we  have 

d 


66  GRAPHICAL  METHODS 


Substituting  the  values  of  Vj,  i?2,  etc.,  and  differentiating,  we 
obtain 


&£»  —  C£»2)  =  0, 
a—  &*i—  czi2) 

a  -  6Xn  -  CXn2)  Xn  =  0, 


—  a—  6a^—  cx£)x£+.  • 
, 

which  may  be  simplified  to  the  equations 

=  0. 
=  0. 
—  0. 

These  are  called  the  "normal  equations,"  from  which  the 
values  of  the  constants  a,  b,  c,  may  be  computed  by  the 
ordinary  methods  of  elimination,  there  being  now  the  same 
number  of  equations  as  unknowns.  It  will  readily  be  seen 
that  the  process  of  substituting  the  values  of  ^y,  2xy, 
2xy2,  etc.,  in  the  normal  equations  and  the  subsequent  so- 
lution of  the  equations  for  the  constants  is  a  tedious  proc- 
ess, the  labor  involved  increasing  rapidly  with  the  number 
of  constants  to  be  determined. 

For  further  details  regarding  the  method  of  Least 
Squares  consult  Bartlett's  The  Method  of  Least  Squares, 
Wright's  Treatise  on  the  Adjustment  of  Observations,  or 
Merriman's  Least  Squares.  For  special  Graphical  Meth- 
ods see  Peddle's  The  Construction  of  Graphical  Charts. 


PART  III. 
SOLUTION  OF  ILLUSTRATIVE  PROBLEMS, 

AND 

PROBLEMS. 


SOLUTION  OF  ILLUSTRATIVE  PROBLEMS. 


Before  proceeding  to  the  numerical  solution  of  a  pre- 
cision problem,  the  student  should  first  decide  the  follow- 
ing questions: — 

First. — Is  the  formula  to  be  discussed  in  the  simplest 
form  for  precision  treatment?  It  frequently  happens,  by 
the  omission  of  certain  terms  the  deviations  in  which 
evidently  produce  a  negligible  effect  on  the  final  result, 
that  an  apparently  complex  formula  can  be  reduced  to 
a  more  convenient  form.  If  it  can  be  reduced  to  a  pro- 
duct function,  this  should  always  be  done. 

Second. — From  a  consideration  of  the  form  of  function 
to  be  discussed,  a  decision  should  be  made  as  to  which 
method  it  is  better  to  employ  in  the  solution;  that  is, 
whether  to  use  the  general  "  deviation  method  "  involving, 
differentiation  of  the  function  or  the  fractional  or  "in- 
spection" method. 

Third. — Having  decided  these  questions,  the  statement 
of  the  problem  should  be  studied;  that  is,  all  given  data 
should  be  systematically  written  down  and  inspected  to 
see  if  they  are  in  the  proper  form  for  applying  the  method 
of  solution  decided  upon.  If  this  is  not  the  case,  numer- 
ical deviations  S  or  A  should  be  changed  over  into  their 

corresponding  fractional   deviations   ;      or   -v^,    or   vice 

m          j\fJL 

versa,  as  the  case  may  be.  Only  after  the  problem  has 
been  consistently  stated  should  the  actual  solution  be 
begun. 

These  general  directions  are  illustrated  below  by  the 
solution  of  several  typical  problems. 


70          SOLUTION  OF  ILLUSTRATIVE  PROBLEMS 

Problem  i.  —  Given  the  following  mean  values  of  the 
weight  of  four  substances  with  their  respective  deviation 
measures:  — 

MI  =  3147.226  gms.      A.D.  =  0.312  gm. 

102  =  100.4211  gms.      reliable  to  0.015  per  cent. 

w3  =  1  .3246  gms.          Probable  error  P.E.  =  0.001  1  gm. 

w4  =  604.279  gms.        reliable  to  1  part  in  5000. 

(a)  Indicate  any  superfluous  figures  in  the  above  meas- 
urements, considering  each  independent  of  the  others. 

Each  quantity  should  be  carried  out  to  two  places  of  un- 
certain figures  as  indicated  by  the  two  significant  figures  in 
its  average  deviation  (Rule  III.,  p.  24).  The  average  devia- 
tion of  each  measurement  should  therefore  be  computed  for 
each  measurement  if  it  is  not  already  given.  Computing 
the  average  deviations  and  applying  Rules  I.,  II.,  and  III.,  it 
will  be  seen  that  the  correct  number  of  figures  to  be  retained 
is  as  follows:  — 

101  =  3147.23  gms.  A.D.  =  di=  0.31  gm. 
w2=  100.421  gms.  A.D.  =  52=  0.015  gm.  as  100  4^  =  0.015. 

J-UU 

w&=  1.3246  gms.     A.D.  =  53  =  0.0013  gm.  as  P.E.  =  0.85  A.D. 
W)=604.28gms.     AJ>.-  «,-  0.12  gm.  a.^'-  JL  . 

(b)  Which  is  the  most  and  which  the  least  precise  of 
these  measurements? 

When  the  quantities  whose  precision  is  to  be  compared  are 
not  of  approximately  equal  magnitude,  their  relative  pre- 
cision is  found  by  comparing  their  fractional  or  percentage 
deviations,  but  not  their  average  deviations  or  probable 
errors.  Hence  with  the  above  data  we  must  compute  the 
fractional  or  percentage  deviation  of  each  of  the  quantities. 


For  wi,      100  —  =  100          =  0.010  per  cent.  ; 


100  —  =  100  =  0.015  per  cent.  ; 

Wz  iOO 


wa,      100    -  =  100        p  =  0.10  per  cent.  ; 

W,      100  ^  =  100^  =  0.020  per  cent. 

Therefore,  the  order  of  precision  is  wit  wz,  w*,  ws.     It  is  to  be 
noted  that,  although  ws  is  weighed  to  a  much  smaller  fraction 


SOLUTION  OF  ILLUSTRATIVE  PROBLEMS  71 

of  a  gram  than  any  of  the  other  quantities,  it  has  by  far  the 
largest  percentage  deviation,  and  is  to  be  regarded,  there- 
fore, as  the  least  precise  measurement. 

(c)  Find  the  sum  of  the  measurements  and  its  devia- 
tion measure,  retaining  the  proper  number  of  significant 
figures  in  the  computation. 

M  =  Wi  -{- w2  +  w3 -\-  w±. 

The  quantity  having  the  largest  A.D.  is  wi,  its  average  de- 
viation being  0.31  gm.  In  the  units  chosen  to  express  the 
measurements,  the  first  and  second  decimal  places  are  un- 
certain. Therefore,  by  Rule  IV.,  page  24,  two  decimal  places 
only  should  be  retained  in  each  of  the  other  quantities  to  be 

added. 

3147.23  gms. 

100.42  gms. 

1.32  gms. 

604.28  gms. 

M=  3853. 25  gms. 

The  resultant  deviation  A  of  th3  sum  M  is 


A!  =  ~—  .  5i  =  61  =  0.31  gm. 

OWl 

Similarly,  A2  =  52  =  0.015  gm.,  which  is  negligible. 
A3  =  63  =  0.0013  gm.,  which  is  negligible. 
A4=64=0.12  gm. 


Therefore,  A  =  vV  +  542  =  Y/olH2  +  0.12*  =  0.33  gm. 

(d)  Find  the  product  of  the  four  quantities  and  its 
precision  measure. 

M  =  WI  .  w2  .  w3  .  w4. 

By  Rule  V.,  page  24,  the  least  precise  factor  is  103,  which 
is  good  to  only  0.10  per  cent.,  and  in  which,  therefore,  five 
significant  figures  should  properly  be  retained  in  a  compu- 
tation, the  last  two  being  uncertain.     Five  figures  should 
likewise  be  retained  in  each  of  the  other  factors,  and  five 
place  logarithms  should  be  used  in  the  computation. 
wi  =3147.2    log  =3.49793 
wz  =  100.42    log  =  2.00182 
tos=l  .3246    log  =  0.  12209 
W*  =  604.28    log  =  2.78124 
log  M=  8.40308 
or  M=  252980000.  gms.4 


72  SOLUTION  OP  ILLUSTRATIVE  PROBLEMS 

The  precision  of  M  should  be  computed  by  the  fractional  or 
inspection  method,  as  it  is  a  product  function.  By  referring 
back  to  problem  (6)  ,  it  will  be  seen  that  the  percentage  pre- 
cision of  w\,  wz,  and  w*  is  between  five  and  ten  times  as  great 
as  that  of  ws.  Hence  practically  all  of  the  uncertainty  in  the 
product  will  result  from  the  deviation  in  this  factor  alone. 
As  M  is  directly  proportioned  to  the  first  power  of  ws, 

100^=  100  6i 

M  ws 

and    therefore    the    percentage    deviation    of    the    product 

100  ~  =  0.10  per  cent.     Hence  A  =  M  x  ~  =  250000.  glnZ4 

(e)  Suppose  the  quantities  are  to  be  combined  by  the 
formula 

M  =  Wi  X  w2—  w$  X  w±. 

Compute  M  and  its  deviation  measure. 

Before  substituting  the  values  of  w  in  the  actual  calcula- 
tion of  M,  it  is  always  well  to  note  the  approximate  value 
of  the  terms  involved.  From  inspection  of  the  data  it  is 
evident  that  wix  wz  =  3  10000.  approximately,  and,  as  the 
least  precise  factor  w%  is  good  to  0.015  per  cent.,  this  product 
should  be  computed  to  six  significant  figures,  of  which  the 
last  two  will  be  uncertain.  The  second  place  of  uncertain 
figures  thus  falls  in  the  units'  place;  anything  beyond  this 
is,  therefore,  negligible.  The  term  ws  X  w\  =  600.  approxi- 
mately, and,  as  this  is  to  be  subtracted  from  wiXwz,  it  is 
useless  to  compute  it  beyond  the  units'  place,  i.e.,  three  sig- 
nificant figures  are  sufficient.  Six  place  logarithms  should 
therefore  be  used  in  computing  w\  x  wz,  and  three  place 
logarithms,  short  multiplication,  or  a  slide  rule  in  computing 

108  X  W4. 

Thus  wi  =  3147.23  gms.  log  w\  =  3.4979284 

wz=  100.421  gms.  log  w2=  2.0018245 

log  wix  wz  =5.4997529 

.'  .wiXw2=  316048.  gms.2 
Ws  =  1.32  gms.  log  ws  =  0.121 

tC4  =  604.  gms.  log  W*  =  2.781 

log  wsXW4=  2.902 

=798.  gms.2 


AT  =316048  gms.2-798  gms?=  315250  gms.2 


SOLUTION  OF  ILLUSTRATIVE  PROBLEMS          73 

To  determine  the  precision  of  M,  we  note  that,  if  the  for- 
mula be  treated  in  the  form  M  =  w\  x  w%  -  wz  X  w*,  we  must  use 
the  general  differential  method  and  find  the  effect  of  each  5 
on  M,  and  take  the  square  root  of  the  sums  of  the  squares. 
It  is  evident,  however,  since  ws  is  good  to  0.10  per  cent., 
and  W4  to  0.02  per  cent.,  that  the  first  three  significant  figures 
of  the  product  ws  .  w±  are  known  exactly,  and,  therefore,  the 
deviations  in  ws  and  w*  introduce  no  uncertainty  in  the  final 
result  M .  The  whole  uncertainty  comes  from  the  measure- 
ments MI  and  v>2.  As  WB  x  w\  is  also  numerically  small  com- 
pared with  wi  x  wz  we  may,  in  the  precision  discussion,  neglect 

it  and  write 

M  =wi  x  W2  approximately, 

and  obtain  the  precision  of  M  by  the  fractional  method. 
The  resultant  fractional  deviation  in  M  is 


M 

But  ~  =  - -  =  0.00010 

M     wi 

and  T?--  =0.00015 

M        102 


Therefore,      =>= 0.0001  J  I2 +  1.52  =  0.00018 


or  100  M  =  0.018  per  cent. 

and  A  =  320000.  gms!2  x  0.00018 = 58  gms? ; 

that  is,  the  value  of  M  is  uncertain  by  ±  58  units. 

The  above  problem  illustrates  the  manner  in  which  the 
number  of  significant  figures  to  be  retained  in  a  measure- 
ment depends  entirely  upon  the  way  in  which  it  enters 
into  the  computation.  Thus  in  (d)  w3  was  required  to 
its  full  precision,  while  in  (e)  it  might  have  been  measured 
much  less  precisely. 

Problem  2. — It  is  desired  to  determine  the  amount  of 
heat  H  generated  in  one  hour  by  a  certain  incandescent 
lamp,  together  with  its  deviation  expressed  in  calories 
and  in  per  cent.  Suppose  mean  measurements  obtained 
by  an  ammeter  and  voltmeter  give  7  =  2.501  ±0.012 
amperes,  and  E  — 109.72  ±  0.34  volts  respectively,  and  the 
time  of  opening  and  closing  the  circuit  is  uncertain  by 


74  SOLUTION  OP  ILLUSTRATIVE  PROBLEMS 

±  0.5  second  at  each  operation.     The  value  of  H  expressed 
in  calories  is 

#  =  0.2390  /.#.£. 

The  expression  for  H  is,  as  it  stands,  a  simple  product 
function  of  the  variables  /,  E,  aiid  t,  and  cannot  be  further 
simplified.     The  problem  should  therefore  be  solved  by  the 
fractional  method. 
The  data  given  are 

7  =2.501  amp.       8f  =  0.012  amp. 
E  =  109.72  volts.    8£  =  0.34  volt. 
t  =  t<2  —  h  =  l  hour  =  3600  sec. 
5^  =  5*2  =  0.50  sec.,  but  dt  is  unknown. 

The  desired  results  are  the  value  of  H,  its  deviation  A  in 

A 

calories,  and  its  percentage  deviation  100  —  . 

H 

The  first  step  is  to  change  the  given  deviations  5  in  each 
component  into  their  respective  fractional  deviations. 


The  fractional  deviation  in  7  is      =  =  0-0048. 


The  fractional  deviation  inE  ia^  =  —~  =  0.0031. 

E      110. 

To  find  the  fractional  deviation  in  t,  we  must  first  compute 
its  numerical  deviation.  Since  t  =  t% — t\  and  5^  =  5<2  =  0.5  sec., 
the  numerical  deviation  in  t  is  5  =  y/rA12+A22; 

dt 
but  AI  =  — •  .5^=5^ 

dt 

and  A2  =  —7  .  ota  ==  8ta 

dt2     2        2 


Therefore          8  =  V^O.52  +  0.52  =  0.70  sec., 
and  the  fractional  deviation  in  t  is 


This  is  seen  to  be  negligible  compared  with  the  fractional 
deviation  in  the  current  7  and  voltage  E  (see  page  31). 
Hence  the  resultant  fractional  deviation  in  H  is 


H 


SOLUTION  OF  ILLUSTRATIVE  PROBLEMS          75 

By  inspection  of  the  formula  for  H,  since  /  and  E  both  enter 
as  first  power  factors, 

^  =  *'  =  0.0048 

£*  =  *£  =  0.0031. 
H        E 

Therefore,   ^  =  V/O00482+  O^OSl2  =  0.0057, 

or  the  percentage  deviation  in  H  is  100  -^  =  0.57  per  cent. 

ti 

In  computing  the  value  of  H,  we  note  that  the  least  pre- 
cise factor  is  the  current  which  is  uncertain  by  0.48  per  cent., 
and  hence  should  be  carried  in  the  computation  to  four  sig- 
nificant figures.  H  should  therefore  be  computed  by  four 
place  logarithms,  four  figures  being  retained  in  each  factor, 
including  the  constant  for  transforming  Joules  to  calories. 
H  =  0.2390  x  2.501  x  109.7  X  3600 

=  236100  calories. 

The  numerical  deviation  A  in  H  is  obtained  at  once  from  its 
fractional  deviation,  as 

A  =  240,000  x  0.0057  =  1400  calories. 

Problem  3.  —  The  candle  power  of  a  gas  flame  is  meas- 
ured against  a  standard  candle  by  means  of  a  photo- 
meter, the  flame  being  placed  at  the  end  of  a  bar  100 
inches  from  the  candle.  Suppose  the  mean  of  a  series 
of  disk  settings  gave  a  =  20.17  ±0.27  inches,  a  being  the 
distance  of  the  disk  from  the  candle.  Compute  the 
candle  power  of  the  flame  and  its  deviation,  assuming 
that  the  candle  is  burning  at  its  normal  rate. 

L    (flame)  _(100-q)2 
L0  (candle)  a* 

/100  —  a\2       /100  —  aV 
This    may    be    written  L  =  L0  /  -  /    =  (  -  / 

as  L0  =  1  candle  power  =  1  c.p.  =  constant. 
100.—  20.17 


20.17 

L  is  a  function  of  a  single  variable  a;  it  is  to  be  noted  in 
the  precision  discussion  that  the  function  cannot  be  regarded  as 
a  fraction  in  which  a  deviation  in  the  numerator  is  inde- 


76  SOLUTION  OF  ILLUSTRATIVE  PROBLEMS 

pendent  of  the  deviation  in  the  denominator.     The  formula 
may,  however,  be  simplified  by  writing  it  in  the  form 

100  -a 

L'=\IL=\X—^- 

and  solving  first  for  the   deviation  in  L',  after  which  the 
desired  deviation  in  L  can  be  Easily  found. 

The  deviation  A'  in  L',  due  to  a  deviation  5  =  0.27  inch  in 

A,      d/lx!00-vA  Ix100    5 

da  \  a      /  a2 

=  l(c.p.)*  =r=  x  0.27  cm.  =  0.068  (c.p.)*. 
20  cm. 

To  find  now  the  deviation  in  L,  we  may  proceed  in  either  of 
two  ways:  — 

First,  General  Method  : 

L'  =  /Z  =  L» 


where  A  is  the  desired  deviation  in  L. 
Therefore,  A  =  2y/L  .  A' 


=  2x  y/16  c.pTx  0.068  (c.p.)i 
=  0.54  c.p. 

Second,  Fractional  Method: 

The  fractional  deviation  in  L'  corresponding  to  A'  is 
A'      0.068      0.068 


and,  since  L'—L^,  the  fractional  deviation  in  L'  is  one-half 
the  fractional  deviation  in  L; 


1  A 

2  '  L'  • 

2x  0.017  =  0.034. 


..  . 

Li 

Therefore,  A  =  0.034  x  L  =  0.034  x  16  c.p.  =  0.54  c.p. 

The  same  result  would  of  course  be  obtained  by  applying 
the  general  differentiation  method  to  the  original  formula 
for  L,  but  the  resulting  value  of  the  differential  coefficient  is 
somewhat  more  complicated. 


SOLUTION    OP   ILLUSTRATIVE    PROBLEMS  77 

On  the  assumption  that  the  average  light  emitted  by 
the  candle  during  the  measurements  is  equal  to  one  candle 
power,  the  candle  power  of  the  gas  flame  is  15.67  +  0.54  c.p.; 

0  54 
i.e.,  it  is  known  to  only  100  -^-  =  3.4  per  cent.,  although  the 

0  27 
original  photometer  setting  a  is  good  to  100  -—  =  1.4  per  cent. 


Problem  4.  —  Suppose  the  index  of  refraction  n  of  a 
substance  is  to  be  determined  by  measuring  the  angle  of 
incidence  i  and  the  angle  of  refraction  r  of  a  ray  of  light. 
If  approximate  measurements  give  z  =  45°  and  r  =  30°, 
how  precisely  should  these  two  angles  be  measured  to 
give  n  to  0.2  per  cent.? 

The  formula  for  n  is 

_sin  i 
'""sin  r* 

As  this  is  not  a  product  function  of  the  variables  i  and  r,  we 
must  use  the  general  deviation  method  if  we  treat  the  for- 
mula in  the  above  form.  If,  however,  we  change  variables 
to  x  and  y,  letting  x  =  sin  i  and  y  =  sin  r,  the  formula  becomes 
n=-,  to  which  we  may  apply  the  fractional  method  of  solu- 

tion for  finding  the  allowable  deviations  in  x  and  y.  Having 
done  this,  however,  we  still  have  two  new  problems  to  solve 
by  the  general  method,  namely,  the  determination  of  the 
deviations  in  i  and  r  from  the  equations  x  =  sin  i  and  y  = 
sin  r.  Both  methods  lead,  of  course,  to  the  same  result. 
We  will  solve  the  problem  both  ways. 
First  Solution.  General  Method: 

sin  i 
n=-  —  . 
sm  r 

Given  100  -  =  0.2;  i=45°;  r  =  30°;  to  find  «,-  and  Sr. 

We  must  first  find  the  value  of  the  A  in  n  from  the  pre- 
scribed percentage  precision  before  proceeding  to  the  solu- 
tion. This  necessitates  knowing  the  approximate  value  of  n, 
which  is  easily  obtained  from  the  data; 

sin  45        1.1 
"  =S30  =  ^  ^  2  =  1A  *PProx,mately. 

Hence  A  =  1.4  X  0.0020  =  0.0028. 


78          SOLUTION  OF  ILLUSTRATIVE  PROBLEMS 

Distributing  thia  deviation  between  i  and  r  by  equal  effects, 

A      0.0028        ^n 
A/=  Ar  =  —  =  —  —7-  =  0.0020. 

Y/2 

dn    ,       cosi    „ 

But  *--" 


.  -  .  „  =  A<  JET  _  0.0020  -          -  0.0014 
cos  i  cos  45 

To  express  this  allowable  deviation  in  the  angle  in  degrees, 
we  note  that 

1°  =     -  =  0.017  radians, 


therefore,  *«•  =  =  0.082°,  or  4.9'. 


dn    ,        sini  cosr 
Similarly,  A,  =  -  .  5r  =      sin2?.       .  «r 


=  0.0020    .  qn  =  0.00082, 

cosr  sin  45  cos  30 

0  00082 
or,  expressed  in  degrees,  ^.~r2f  =0.048°,  or  2.9'. 


The  solution  shows,  therefore,  that,  to  obtain  a  precision 
of  0.2  per  cent,  in  n,  an  instrument  should  be  used  capable  of 
reading  to  at  least  3'.  In  practice  one  graduated  to  read  to 
minutes  would  be  chosen. 

Second  Solution.  Fractional  Method:  If  we  put  z=sin  i 
and  y=sin  r,  then  n=-,  and  we  may  use  the  inspection 
method  as  follows.  The  prescribed  fractional  deviation  is 
stated  to  be  not  greater  than  -  =  0.0020.  Hence,  distribut- 
ing this  deviation  between  x  and  y  by  equal  effects,  we  have 


n       n      y/2     n     y2 
But  by  inspection  of  n  =-  it  is  seen  that 


Hence  -  =  0.0014  and  ^  =  0.0014. 

We  have  now  two  new  problems  to  solve,  namely,  to  find 
8t  and  5r  from  the  above  values  of  the  allowable  precision  in 


SOLUTION   OF  ILLUSTRATIVE   PROBLEMS  79 

x  and  y.  As  z=sin  i  is  a  trigonometric  function,  we  must 
go  back  to  the  general  differentiation  method  to  find  the 
deviation  in  i  corresponding  to  a  deviation  &x  in  x. 

As  ^  =  0.0014, 

we  have         8*  =  0.0014  x  =  0.0014  sin  45°  =  0.0010. 

d  sin  i 


di 


cos  i . 


6,=-       =  0.0010  -r  -      =0.0014, 

cos  i  V2 


or  in  degrees  5,  =  =  0.082°  =  4.9'. 


Similarly,  to  find  5r,  given  y  =  sin  r  and  —  =  0.0014,  we 

have 

dy  =  0.0014  y  =  0.0014  sin  30°  =  0.00070. 

Also  5,=  —        .  5r=  cos  r  .  5r 

-a00070  *     - 


or  in  degrees  6,  =  =  0.048°  =  2.9'. 


In  this  problem  there  is  evidently  no  saving  of  labor  by 
transforming  the  function  to  the  product  form  and  first 
using  the  fractional  method,  as  the  ultimate  solution  necessi- 
tates going  back  to  the  general  method. 


PROBLEMS 


Questions  and  Problems. 

1.  Explain  the  terms:   precision  measure;  deviation  measure; 
constant  error;  residual  error;   probable  error;  mean  error;  huge 
error;  indeterminate  error;  weighted  mean;  weights. 

2.  What   is   the   geometrical    significance   of   the    average 
deviation,  probable  error,  and  mean  error  in  relation  to  the 
curve  representing  the  law  of  chance? 

3.  Is  it  practicable  to  reduce  the  average  deviation  of  a 
mean  result  to  any  desired  value  by  increasing  the  number 
of  observations?     Why? 

4.  If  the  mean  value  of  the  length  of  a  rod  computed  from 
nine  measurements  is  24.213  cm.  A.D.  =  0.012  cm.,  how  many 
more  similar  observations  should  be  made  in  order  that  the 
A.D.  =0.0060  cm.? 

5.  Under  what  circumstances  may  an  observation  properly 
be  rejected,  and  why? 

6.  What   determines   the   number  of  places   of  significant 
figures  to  be  retained  at  any  part  of  a  computation? 

Under  what  circumstances  should  four,  five,  or  seven-place 
logarithms  be  used  in  a  computation? 

7.  Do  the  number  of  significant  figures  in  a  result  depend 
upon  the  position  of  the  decimal  point?     Explain  reasons  for 
your  answer.     Does  the  precision  of  a  result  depend  upon  the 
position  of  the  decimal  point?     Why? 

8.  Explain    why    when    adding     or    subtracting    observed 
quantities   we   are   governed  by   decimal  places  in  rejecting 
figures,  but,  when  multiplying  or  dividing,  places  of  signifi- 
cant figures,  regardless  of  the  decimal  point,  must  be  con- 
sidered. 


82  PROBLEMS 

9.  Given  the  following  measurements  and  their  deviation 
measures: — 

241.631  gms.  reliable  to  0.25  per  cent. 

1620.124  gms.  A.D.  =  0.81  gms. 

10.005  gms.  reliable  to,  1  part  in  1,000 

7141.110  gms.  P.E.  =  ti.603  gms. 

(a)  Indicate  all  superfluous  figures. 

(6)  Which  is  the  most  and  which  the  least  precise  quantity, 
and  why? 

(c)  How  many  figures  should  be  retained  in  each  quantity 
in  computing  their  sum? 

(d)  How  many  figures  should  be  retained  in  each  quantity 
in  computing  their  product? 

(e)  How  many  figures  should  be  retained  in  each  quantity 
in  computing  the  difference  between  the  product  of  first  two 
and  the  product  of  the  last  two  quantities? 

10.  Given  the  following  measurements  of  the  length  of  a 
rod  and  their  precision  measures: — 

24.316    cm.  A.D.  =  0.028  cm. 

24.3922  cm.  fractional  precision  12  parts  in  1,000 

24.358    cm.  P.E.  =  0.0121  cm. 

24.3091  cm.  reliable  to  0.11% 

24.3100  cm.  A.D.  —  0.0172  cm. 

(a)  Indicate  any  superfluous  figures. 

(6)  Indicate  the  order  of  reliability  of  the  results. 

(c)  Compute  the  relative  "weights." 

(d)  Compute  the  weighted  mean. 

11.  The  precision  measures  of  four  independent  determina- 
tions of  the  modulus  of  elasticity  of   steel  are  expressed  as 
follows:    1st,   0.60    per    cent.;    2d,    2    parts    in    1,000;    3d, 

probable  error  =  4.2     ^m"-  4th,  average  deviation  =  10.     gm'- 
mm.  mm. 

The  modulus  of  elasticity  is  about  20,000         2'- 

171171. 

Which  measurement  is  the  most  reliable?    Find  the  rela- 
tive weights  of  the  four  determinations. 


PROBLEMS  83 

12.  Independent  determinations  of  the  rate  of  a  pendulum 
by  different  methods  and  observers  gave 

0.70061  sec.  A.D.      =  0.00023  sec. 
0.70047    "    A.D.      =0.00069    " 
0.70056    "     correct  to  0.092% 
0.70051    "    P.E.      =  0.00039  sec. 

(a)  Which  is  the  most  reliable  observation? 
(6)  Compute  the  relative  "weights"  of  the  observations. 
(c)  Indicate  how  to  compute  the  best  representative  value 
of  all  the  observations. 

13.  The  dimensions  of  a  right  cylinder  are  found  to  be  as 
follows:  — 

length      =  12.183  cm.     A.D.  =  0.024  cm. 
diameter  =    4.242  cm.     A.D.  =  0.021  cm. 

Find  the  volume  and  its  deviation  measure,  indicating 
proper  number  of  significant  figures  at  each  step  of  the  com- 
putation. 

14.  The  diameter  of  a  spherical  globe  is  found  to  be  approxi- 
mately six  inches.     If  the  average  diameter  varies  by  0.1  per 
cent.,  what  variation  in  cubic  inches  will  this  produce  in  the 
volume?     If  the  variation  in  the  diameter  is  0.0020  inch,  what 
uncertainty  will  result  in  the  volume? 

15.  The   length  of  a  physical  pendulum   is  given   by  the 
expression 


where  r  =  \  diameter  of  ball  =  -, 

and  h  =  distance  of  knife  edge  from  the  top  of  the  ball. 

Suppose  h  =  100.031  cm.  A.D.  =  0.027  cm. 

d=      6.256cm.  A.D.  =  0.022  cm. 

(a)  Find  the  resultant  deviation  in  I. 

(b)  How  many  significant   figures  should  be  retained  in 
each  term  of  the  formula,  in  computing  I,  and  why  ? 

(c)  Is  it   necessary  to    consider  the  third   term  in  this 
formula  in  a  precision  discussion  of  If     Why? 


84  PROBLEMS 

16.  The  mean  time  of  nine    coincidences  of  two  beating 
pendulums     is     50.43     seconds.     A.D.  =  0.11     second.    The 
standard  pendulum  beating  true  seconds   gains  on  the  other 
pendulum.     Compute  the  true  time  of  vibration  of  the  latter 
and  its  precision  measure. 

17.  The  time  of  swing  of  a  half-second  pendulum  is  measured 
to  0.20  per  cent.     The  length  is  measured  to  0.10  mm.     Find 
the  precision  of  the  computed  value  of  g. 

18.  Measurements  with  a  spherometer  gave  for  a   certain 

lens 

h=    1.22110mm.     A.D.  =  0.00088  mm. 
r  =  35.735  mm.        A.D.  =  0.061  mm. 


Compute  R,  the  radius  of  curvature  of  the  lens  retaining 
the  proper  number  of  significant  figures  throughout  compu- 
tation. What  deviation  would  be  introduced  in  R  by  a  devi- 
ation of  0.00088  mm.  in  h  ?  By  a  deviation  of  0.061  mm.  in  r  f 
What  would  be  the  combined  effect  of  these  deviations  on 

R?  Is  the  term  —  negligible  in  the  above  precision  discus- 
sion, and  if  so,  why? 

19.  A  10  ohm  coil  is  standard  at  15°  C.  What  will  be  its 
resistance  at  30°  C.  and  how  precise  will  this  be  known  if  the 
temperature  is  determined  to  0.1  degree? 


-4-  0.00388  (t  —  15)1 . 


How  closely  must  t  be  known  in  order  that  the  resistance 
may  be  depended  upon  to  0.02  per  cent.? 

20.  The  electromotive  force  of  a  Clark  cell  is 
Et  =  1.4340  [~1  —  0.00078  (t  —  15)1 . 

How  closely  must  t  be  measured  in  order  that  E  may  be 
known  to  0.05  per  cent.? 


PROBLEMS  00 

21.  Given  a  coil  of  wire  the  resistance  of  which  at  15°  C.  may 
be  assumed  to  be  exactly  100  ohms.    The  wire  is  an  alloy  the 
resistance  of  which  at  t°  may  be  computed  by  the  expression 

Rt=Ris  fl  +  0.00051  (t— 15)1. 

Calculate  the  amount  of  heat  H  generated  in  the  wire  in  an 
hour's  run  by  a  current  of  11.273  amperes,  if  the  wire  is  main- 
tained at  a  temperature  of  45°  C.  H=I2Rt. 

If  the  temperature  is  known  to  ±1.0°,  the  duration  of  the 
run  to  ±1.0  second,  and  the  current  to  ±0.011  ampere,  calcu- 
late the  deviation  in  H  in  Joules  and  in  per  cent.  Are  the 
deviations  in  any  measurements  negligible,  and  if  so,  why? 

22.  The  sensitiveness  of  a  spirit  level  is  0  =  y  X  206265". 

(a)  If  I  =  380.  mm.  approx.  and  h  =  0.0597  mm.  ±  0.0018 
mm.,  how  many  figures  would  you  keep  in  the  value  01 

(b)  How  many  would  you  use  in  the  value  of  the  radian, 
206265?     How  many  in  h? 

(c)  If  I  and  206265  are  to  introduce  no  deviation  in  the  re- 
sult in  comparison  with  h,  what  is  the  allowable  percentage 
deviation,  and  how  few  figures  may  be  used  in  each  ? 

(d)  If  you  wished  to   determine   0  with   greater  precision 
than  above,  in  which  measurement  would  you  use  greater 
care? 

23.  In  calibrating  a  burette  by  drawing  off  water  for  each 
10  cc.  and  weighing  it  in  a  flask,  how  precise  should  the  weigh- 
ings be  made  if  the  calibration  is  to  be  reliable  to  0.01  cc.? 
Suppose  the  calibration  were  made  with  mercury,  how  close 
should  the  weighings  be  made?     Is  it  necessary  to  note  the 
temperature  of  the  water  in  this  calibration  and  why? 

24.  It  is  desired  to  calibrate  a  flask  to  hold  1,000  cc.  at 
20°  C.,  the  calibration  to  be  reliable  to  0.5  cc.     What  weight 
(using  brass  weights  sp.   gr.  =  8.5)   would  you  add  to  the 
weight  of  the  flask  empty,  so  that  it  would  exactly  balance 
the  water  having  the  desired  volume?     How  precise  would 
you  make  your  weighings?     Would  it  be  necessary  to  take 


86  PROBLEMS 

into  account  the  barometer  reading  in  figuring  the  correc- 
tion for  reduction  to  vacuo?     Why? 

25.  The  per  cent,  of  silver  in  a  certain  alloy  is  determined 
gravimetric  ally  by  weighing  the  amount  of  silver  present  as 
silver    chloride.     Suppose    an  -''analysis    gave    the    following 
results: — 

Wt.  of  alloy    1.43252  gms.     ±  0.00014  gm. 
"     "AgCl    0.19513  gms.     ±0.00021  gms. 

Compute  the  per  cent,  of  silver  in  the  alloy  and  the  preci- 
sion with  which  this  would  be  known. 

Atomic  weight  silver      ==  107.93  ±  0.02 
Atomic  weight  chlorine  =    35.45  ±  0.03 

26.  A  specific  gravity  determination  by  Archimedes'  prin- 
ciple gave  the  following  results: — 

weight  of  substance  in  air  =  10.2431  gms.  ±  .0004  gm. 
weight  of  substance  in  distilled  water  at  20°  C.  =  9.0422 

gms.  ±  .0010  gm. 
density  of  water  at  20°  C.  =  0.99825. 

Compute  specific  gravity  and  its  precision  measure,  using 
the  correct  number  of  places  of  significant  figures  through- 
out the  computation.  Is  the  correction  for  reduction  of 
weighings  to  vacuo  negligible  in  this  case,  and  why? 

27.  Given 

wt  =  (Ww  -  6)  [l  +  k  (t°  -  20°)]-— 

where   w2o  =  27.6231    grams  =  weight  of  bottle   plus  water. 
b  =  12.6193  grams  =  weight  of  bottle. 
k  =   0.000026  per  degree  C. 

Dt  and  D2o  =  density  of  water  at  t°  and  20°  respectively. 

The  density  D  diminishes  0.02  per  cent,  per  degree  C. 
What  is  the  greatest  allowable  value  which  t  may  have  and 
the  correction  term  due  to  expansion  of  glass  be  negligible; 
i.e.,  affecting  the  value  of  (WZQ  —  b)  by  less  than  0.0001  gm.? 
Is  the  correction  due  to  change  of  density  negligible  for  t  =  22° 


PROBLEMS  87 

and  why?  If  D20  =  0.99827  and  t  =  25°,  compute  wt,  using 
as  few  figures  as  practicable. 

28.  With  what  precision  would  it  be  necessary  to  meas- 
ure the  length  and  diameter  of  a  right  cylindrical  column 
approximately  12  inches  long  and  6  inches  in  diameter   in 
order   to   determine   the   volume  to   0.10   per   cent.?     What 
deviation  in  cubic  inches  would  this  correspond  to? 

29.  If  it  is  desired  to  compute  the  area  of  a  circle  approxi- 
mately 10  sq.  cm.  in  area  to  5  parts  in  10,000,  how  precise 
should  the  diameter  be  known?     How  many  places  should  be 
retained  in  IT  in  the  computation? 

30.  If  the  volume  of  a  sphere  is  computed  from  a  measure- 
ment of  its  diameter,  how  precise  should  the  latter  be  measured 
in  order  that  the  former  may  be  reliable  to  0.1  per  cent.;  to  1 
part  in  500? 

If  the  sphere  is  approximately  six  inches  in  diameter,  what 
precision  in  the  diameter  does  a  precision  of  0.1  per  cent,  in  the 
volume  require?  If  the  A.D.  of  the  diameter  is  0.022  inch, 
what  is  the  precision  of  the  volume? 

31.  The  ratio  of  the  length  of  the  arms  of  a  balance  is  given 
by  the  expression 

_  length  right  arm */Wi 

length  left  arm         VW7' 

where  Wi  and  Wr  are  the  observed  weights  of  a  given  mass 
when  weighed  in  the  left  and  right  hand  pan,  respectively.  If 
the  mass  weighs  approximately  20  grams,  with  what  precision 
must  Wi  and  Wr  be  determined  in  order  that  the  ratio  r  may 
be  reliable  to  0.01  per  cent.? 

32.  The  specific  gravity  of  a  platinum  alloy  is  desired  to 
0.1  per  cent.     The  method  based  on  Archimedes'  principle  is 
to  be  used.     The  weight  of  substance  in  air  is  about  42  grams. 
The  specific  gravity  is  approximately  21.     To  what  fraction 
of  a  gram  should  each  weighing  be  made?     What  corrections 
should  be  applied? 


88  PROBLEMS 

33.  A   freely   falling    body    passes    through    the    distance 
h=  %gt2  in  t  seconds.     It  is  desired   to  determine  g  to  one- 
tenth  per  cent,  from  a  measurement  of  h  and  t. 

(a)  How  precisely  should  h  and  t  be  measured? 

(b)  If  h  =  4  meters,  what  will  be  the  allowable  numerical 
deviation  in  h  and  t? 

(c)  If  h  —  4  meters  and  it  is  found  that  Bh  =  1.0  mm.  and 
8*  =  0.0014  second,  what  will  be  the  resultant  deviation  in  gf 

34.  If  g  is  to  be  computed  from  the  mean  of  a  series  of  nine 
observations  on  the  time  of  swing  of  a  pendulum,  the  length  of 
which  is  I  =  1  meter,  what  must  be  the  percentage  precision 
of  t  and  I  in  order  that  g  be  precise  to  0.1  per  cent.?     What 
will  be  the  value  of  ad.  and  A.D.  in  the  case  of  t?     How  many 
figures  should  be  retained  in  IT? 

35.  The  gas  constant  R  is  given  by  the  formula 

R  =  2?  where  T  =  t°  +  273°. 

(a)  How  precisely  must  p,  v,  and  t  be  measured  in  order 
that  R  may  be  reliable  to  0.1  per  cent.? 

(6)  How  large  a  deviation  will  be  introduced  in  R  by  a 
variation  of  1°  C.  in  the  temperature  alone  at  20°  C.  ? 

36.  The  indicated  horse  power  (I.  H.  P.)  of   an  engine  is 

P  X  L  XA  X  N 
given  by  the  expression  7.  H.  P.  = . 

Suppose  for  a  given  engine  the  approximate  values  of  these 
quantities  are  as  follows: — 

P  =  50  Ibs.  per  sq.  in.  =  mean  effective  pressure. 

L  =  2  feet  ==  length  of  stroke. 

A  =  Area  of  piston,  the  diameter,  D,  of  which  is  16  inches. 

N  =  100  =  number  of  strokes  per  minute. 

How,  precisely,  should  P,  L,  D,  and  N  be  determined  in  order 
that  the  computed  horse  power  of  the  engine  may  be  reliable 
to  1  per  cent.?  To  one-quarter  of  a  horse  power? 

37.  A  rectangular  steel  rod  of  breadth  b  and  depth  d  is 
supported  at  its  ends  and  loaded  at  its  centre  by  a  weight  W. 


PROBLEMS  89 

If  I  is  the  length  of  the  rod  between  its  supports  and  a  is 
the  deflection  at  the  centre, 

WP 
4Ebd3  ' 

where  E  is  the  modulus  of  elasticity. 

Suppose  measurements  of  these  quantities  gave 

b=    8.113mm.  8b  =  0.042  mm. 

d  =  10.50    mm.  8d  =  0.025  mm. 

I  =    1.000  meter  precise  to  one  part  in  5,000 

a  =    2.622  mm.  precise  to  0.25% 

W  =    2  kgms.  precise  to  0.02  gram 

Compute 

(a)  E,  the  modulus  of  elasticity. 

(6)  The  deviation  in  E  due  to  each  component. 

(c)  The  resultant  deviation  in  E. 

38.  What  would  be  the  allowable  deviations  in  the  meas- 
urement of  b,  d,  I,  and  a  for  the  beam  defined  in  problem  37 
if  the  value  of  E  is  to  be  reliable  to  \  per  cent.?     Assume 
the  error  in  the  value  of  W  to  be  negligible. 

Do  you  think  this  precision    could    be  readily    attained? 
Why? 

39.  The  modulus  of  elasticity  E  of  a  cylindrical  wire,  of 
length  I,  cross  section  q,  which  when  loaded  with  a  weight 
w  is  elongated  by  an  amount  a,  is  given  by  the  expression: 

£  =  ^. 
aq 

a  =  mi  —  m0  where  mi  and  m0  are  mean  micrometer  read- 
ings when  the  wire  is  under  a  load  of  w  kilograms  and  no 
load,  respectively,  q  =  \trd?  where  d  is  the  mean  diameter 
of  the  wire. 

Given  the  following  data: 

I  =  200.11  cm.  a.d.  =  0.05  cm. 

w  =  10  kilograms  accurate  to  1  gram. 

mi  =  9.4255  mm.  A.D.  =  0.0024  mm. 

m0  =  8.2233  mm.  A.D.  =  0.0012  mm. 

d  =  1.002  mm.  correct  to  0.2% 


90 


PROBLEMS 


(a)  Compute  the  average  deviation  and  the  percentage  de- 
viation of  q. 

(b)  Compute  the  deviation  of  a. 

(c)  Compute  E,  using  proper  number  of  significant  figures. 

(d)  Compute  resultant  percentage  error  of  E. 

(e)  Are  any  of  the  above  dat^  more  precise  than  necessary 
and  why  ? 

40.  (a)  If  it  is  desired  to  determine  E  for  the  above  sample 
of  wire  to  0.5  per  cent.,   how  precisely  should  the  compo- 
nents a  and  q  be  measured,  assuming  that  the  deviations  in 
I  and  w  can  be  made  negligible? 

(b)  How  precisely  must  d  be  measured  to  fulfil  this  condi- 
tion? 

(c)  How  precisely  must  Wi  and  m0  be  measured? 

(d)  Do  you  think  this  degree  of  precision  could  be  readily 
attained,  and  why? 

41.  It  is  desired  to  determine  the  focal  length  of  a  plano- 
convex  lens  from  spherometer  measurements,    n  »•  1.5  ap- 
proximately. 

.    1 


(a)  If  F  is  desired  to  0.1  per  cent.,  how  precisely  must  n 
and  R  be  determined? 

(b)  If    preliminary    measurements    give  r  =  40  mm.    and 
h  =  4    mm.,  approximately,  how  precisely  should   r   and   h 
be  determined  to  fulfil  condition  a? 

42    The  formula  for  computing  the  wave  length  by  means  of 
a  diffraction  grating  is  for  second  order  spectra  X  =  —  .  sin  0. 

£ifl 

If  the  grating  is  ruled  17,296  lines  to  the  inch,  and  a  prelimi- 
nary measurement  gives  0  =  53°  approximately  for  the  sodium 
band,  with  what  precision  must  0  be  measured,  and  how 
should  the  optical  circle  be  graduated  to  give  X  to  one  part 
in  10,000? 


PROBLEMS  91 

43.  /  =  K  tan  $  for  a  tangent  galvanometer,  whose  galva- 
nometer constant  is 

K  =  1.963  ±0.002. 

If  the  deviation  in  reading  any  deflection  0  is  8$  =  0.1°,  com- 
pute the  value  of  /  and  its  deviation  measure  for  0  =  45°  and 
0  =  60°. 

44.  The  heating  effect  H  of  an  electric  current  is  to  be 
calculated  from  measurements  of  7,  t,  and  E,  or  7,  t,  and  R. 

#=0.23907^;  77  =  0.2390  PRt. 

(a)  If  H  is  to  be  reliable  to  0.2  per  cent.,  what  must  be  the 
percentage  precision  of  each  of  the  component  measurements 
in  both  cases?     The  allowable  deviation  in  each  measurement? 

(b)  If  E=  110  volts   and  is  measured  to  0.5  per  cent., 
R  =  100  ohms  and  is  reliable  to   1  part  in  1,000,   81  =  0.03 
amp.  and  8t  =  0.2  sec.;  compute  the  precision  of  a  ten-minute 
run  by  each  method,  and  state  which  method  you  consider  the 
better. 

45  Given  the  following   approximate  values  for  a  specific 
heat  determination:  — 

s  =  sp.  ht.  of  substance 
w  =  wt.  of  substance         =  300  grams 
w0  =  "    "   water  =  500  grams 

Wi  =  "    "    calorimeter       =  100  grams 
«i  =  sp.  ht.  of  calorimeter  =  0.095 
(ts  —  t*)  =  fall  of  temperature  of  substance 

=  100°  -  20°  =  80°  C. 
(tz  —  ti  =  rise  of  temperature  of  calorimeter 

plus  water  =  20°  -  15°  =  5°  C. 
(WQ  +  k)  (I*  - 


7:  — 
w  (t,  - 


,  where  k  =  wl 


If  s  is  desired  to  0.5  per  cent.,  with  what  precision  should 
each  of  the  quantities  w,  w0,  Wi,  Si,  tf,  ti,  and  22,  be  measured 
or  known? 

Can  the  deviations  in  any  of  the  above  quantities  be  readily 
made  negligible?  Solve  the  problem  under  these  conditions. 


92  PEOBLEMS 

46.  The  resistance  of  a  metal  bar  is  determined  by  measur- 
ing the  drop  in  potential  between  its  ends  and  the  correspond- 
ing current  flowing  through  it.     Suppose  mean  measurements 
gave 

I  =  11.431  amperes  ±  0.022  ampere, 
E=  0.5073  volt  ±J).010  volt. 

Compute  the  resistance  of  the  bar  and  its  deviation  in  ohms. 

47.  Suppose  the  mean  of  nine  settings  of  a  photometer  disk 
gave 

a  =  240.1  cm.  ±1.1  cm., 

a  being  the  distance  of  the  disk  from  a  gas  flame  whose  candle 
power,  L,  is  desired,  and  300 — a,  the  distance  of  the  disk 
from  a  standard  candle.  Assuming  the  candle  to  be  burning 
at  its  normal  rate,  compute  the  candle  power  of  the  flame  and 
its  percentage  deviation. 

L  (flame)  a2 

V  (candle) "(300— a)2' 

48.  If  E  =  K  loge  — ,  compute  the  deviation  in  E  due  to  a 

Pa 

deviation  of  0.1  per  cent,  in  the  measurement  of  p\  and  pz 
respectively,  K  being  a  constant.  If  the  deviations  in  pi  and 
Pa  were  known  to  be  of  the  same  sign,  what  would  be  their 
resultant  effect  on  E? 

49.  Suppose  a  stop  watch  loses  regularly  at  the  rate  of  1.2 
seconds  in  15  minutes  and  the  uncertainty  of  stopping  and 
starting  the  second  hand  is  ±0.1  second  in  each  case.     If  a 
runner  makes  a  one-mile  record  in  4  minutes  35f  seconds  by 
the  watch,  calculate  the  true  time  and  its  deviation  measure 
in  seconds  and  per  cent. 

50.  To  what  fraction  of  a  gram  should  a  piece  of  aluminum 
be  weighed  in  air  and  in  water,  respectively,  in  order  that  its 
computed  specific  gravity  may  be  reliable  to  one  part  in  a 
thousand?    The  approximate  weight  of  the  sample  in  air  is 
25  grams,  and  its  specific  gravity  is  approximately  2.7. 

51.  Four  independent  observers  determine  a  resistance  of 
about  one  ohm  to  0.10  per  cent.,  0.0030  ohm,  one  part  in  five 


PROBLEMS  93 

hundred,  and  with  a  probable  error  of  0.00085  ohm,  respectively. 
What  are  the  relative  reliabilities  of  the  determinations  and 
their  respective  weights? 

52.  A  certain  32  c.p.  lamp  takes  a  current  of  approximately 
1  ampere  at  110  volts.     If  its  resistance  under  these  conditions 
is  desired  to  0.5  ohm,  how  precise  should   the  current  and 
voltage  be  measured? 

53.  The  mean  of  sixteen  comparisons  of  a  yard  scale  and  a 
standard  meter  scale  gave  the  result: 

1  yard  =  0.91449  m.;    A .D.=  0.00011  m. 

If  there  is  a  residual  error  of  ±0.007  cm.  in  the  meter  scale 
after  all  sources  of  constant  error  have  been  corrected  for,  to 
what  fraction  of  an  inch  is  the  value  of  the  yard  reliable?  If  the 
meter  scale  were  correct  at  0°  C.  and  used  at  20°  C.,  how  large 
a  constant  error  (in  inches)  would  result  if  its  expansion  were 
neglected,  assuming  that  it  expanded  0.0025  per  cent,  per 
degree? 

54.  Two  tuning  forks  of  slightly  different  pitch  "beat"  with 
each  other  45.31  ±  0.45  times  per  minute  when  sounded  simul- 
taneously.    If  the  standard  fork  makes  exactly  256  vibrations 
per  second,  what  is  the  rate  of  the  second  fork  and  its  deviation 
measure,  assuming  it  to  be  of  lower  pitch  than  the  standard? 

55.  Given  the  following  data  on  the  specific  gravity  of  a  sub- 
stance lighter  than  water. 

Weight  of  substance  in  air  =  10.1321  gms.  ±  0.0002  gm. 

Weight  of  substance  and  sinker 

immersed  in  water  at  20.0°  C.  =  8.4418  gms.  ±0.0020  gm. 
Weight  of  sinker  immersed  in 

water  at  20.0°  C.  =  10.4522  gms.  ±  0.0010  gm. 

Compute  the  specific  gravity  of  the  substance  referred  to  water 
at  20.0°  C.  and  its  numerical  and  percentage  deviation.  If  the 
density  of  water  decreases  0.18%  between  4°  C.  and  20°  C., 
compute  the  specific  gravity  referred  to  water  at  4°  C.  How 
precisely  should  the  temperature  of  the  water  at  20°  be  deter- 
mined if  this  last  reduction  is  to  be  reliable  to  0.05%? 


94  PROBLEMS 

56.  Suppose  a  pendulum  approximately  550  feet  in  length  is 
swung  from  the  top  of  the  Washington  Monument.    How  pre- 
cisely should  the  length  and  time  of  vibration  be  measured  in 
inches  and  seconds,  respectively,  if  the  value  of  g  computed 

from  these  data  is  to  be  reliable  -to  0.50  7 -r^  ? 

(second)2 

57.  Suppose  that  fifty  16  c.p.  incandescent  lamps  are  used  on 
the  average  2  hours  a  day  for  4  weeks.     Each  lamp  takes  0.5 
ampere  at  110  volts.    Calculate  the  total  amount  of  energy 
consumed  in  Joules.    If  this  energy  is  measured  by  determin- 
ing the  average  current  and  voltage  by  an  ammeter  and  volt- 
meter each  of  which  reads  uniformly  2%  too  high,  how  much 
overcharge  would  there  be  on  the  lighting  bill  if  the  cost  of 
electrical  energy  is  10  cents  per  kilowatt  hour? 

If  the  average  voltage  and  current  are  uncertain  by  ±1.1 
volts  and  ±0.0050  ampere,  respectively,  what  uncertainty 
would  there  be  in  the  total  electrical  energy  consumed,  and  ia 
its  value  in  dollars  and  cents? 

58.  If  the  weight  of  a  substance  in  air  is 

w  =  49.7631  ±0.0012  grams 

and  it  is  desired  to  calculate  its  weight  W  in  vacuo,  how  closely 
should  the  density  8  of  the  substance,  the  density  A  of  the 
balance  weights,  and  the  density  <r  of  the  air  in  the  balance 
case  be  known  in  order  that  the  correction  to  be  added  to  w  may 
be  computed  to  the  nearest  0.0005  gram? 

Given  8  =  0.8,  A  =  8.4,  and  <r  =  0.0012  approximately. 


59.  The  formula  for  calculating  the  temperature  t  of  a  mer- 
curial thermometer  when  corrected  for  stem  exposure  is 

t  =  ti  +  0.000156  (ti—ta)n, 
where     ti  =  observed  temperature, 

ta  =  temperature  of  exposed  stem, 
n  =  number  of  degrees  exposed  at  ta°. 
Suppose        ti  =  330°  ±  0.5°,        ta  =  30°  ±  0.5°,        and 

n  =  200°  ±  5°    approximately. 
Compute  t  and  its  deviation  measure. 


PROBLEMS  95 

60.  The  formula  for  the  latent  heat  of  vaporization  is 


w 

If  the  rise  in  temperature  tz—  /1  =  25°—  15°  =  10°,  the  fall  in 
temperature  of  steam  ts—  £2  =  100°—  25°  =  75°,  the  condensed 
steam  =  20  grams,  and  the  water  equivalent  wo  +  k  =  1,200 
grams  approximately,  calculate  how  precisely  you  would  de- 
termine each  of  these  four  factors  if  r  is  to  be  reliable  to  0.5%. 
(Assume  r=  540  cal.  for  steam.) 

rr 

61.  If  I  =  -£  —        —  -,  what  is  the  allowable  deviation  in 

-tti  + 


ohms  in  Ri,  Rt,  and  Rs,  respectively,  if  /  is  to  be  reliable  to 
0.1%  and  the  deviation  in  E  is  negligible?  The  approximate 
values  of  the  resistances  are  Ri  =  10  ohms,  and  R2  =  100  ohms, 
and  #3  =  1  ohm.  If  the  above  formula  were 


what  would  be  the  solution  under  the  same  conditions? 

62.  Suppose  Ii=Ki  sin  0  and  h=K*  tan  0.    The  value  of 
Ki  and  K*  are  each   known  to   0.1%;    0  =  45°;    8d  =  ±Q.l°. 
Compare  the  precision  of  /  as  calculated  by  the  two  formulae. 

63.  Suppose  you  had  three  areas,  each  equal  approximately 
to  10  sq.  in.,  one  being  a  circle,  one  a  square,  and  one  an 
equilateral  triangle.    How  precisely  would  you  have  to  know 
the  diameter  of  the  first  and  the  average  value  of  one  side  of 
the  last  two,  in  order  that  the  computed  area  may  be  reliable 
to  0.01  sq.  in.? 

64.  The  capacity  of  a  spherical  condenser  is  given  by  the 


/i  01 

expression  C=—   —  .     Suppose 

n  =  10.0010  cms.  ±  0.0019  cm.; 

rz  =  15.0000  cms.  ±  0.0044  cm.  ; 

K=  2.0130  ±0.0012. 

Calculate  C  and  its  resultant  deviation  measure;  the  percentage 
deviation  in  the  numerator;  the  deviation  measure  of  the  de- 
nominator expressed  in  centimeters. 


96  PROBLEMS 

65.  Given  a  triangular  lot  of  land  whose  three  sides  are  a,  b,  c, 
respectively,  a  =  120  feet  approximately;  b  =  180  feet  approx- 
imately. The  angle  0  between  a  and  6  is  about  45°.  It  is 
desired  to  find  the  area  of  the  triangle  to  0.12%,  How  pre- 
cisely should  a,  b,  and  0  be  measured?  If  a,  b,  and  B  are  meas- 
ured with  the  above  precision,  what  would  be  the  precision  of 
the  computed  value  of  c  ? 

Area  =  \ab  sin  6 


66.  A  sample  of  sodium  chloride,  NaCl,  is  analyzed  by  pre- 
cipitating with  silver  nitrate  and  weighing  the  silver  chloride, 
AgCl. 

Wt.  of  sample  of  NaCl  =  0.5017  gm.  ±  0.0005  gm. 
Wt.  of  AgCl  =  1.1817  gms.  ±  0.0012  gm. 

Assuming  the  atomic  weight  of  sodium  and  chlorine  to  be 
known  to  ±0.1  per  cent,  and  that  of  silver  to  be  known  to 
±0.03  per  cent.,  calculate  the  per  cent,  of  chlorine  present  in 
the  sample  and  the  precision  with  which  you  can  depend  upon 
this  result. 

Atomic  weights: 

Na  =  23.00        01  =  35.46        Ag=  107.88 

67.  The  index  of  refraction  of  a  substance  is  given  by  the 

sim 

expression  n  =  -  —  • 
smr 

If  i  =  45°  ±10'  and  r  =  30°±5',  compute  the  value  of  n  and 
its  average  and  percentage  deviation. 

68.  The  index  of  refraction  of  a  substance  as  determined  by 
the  Pulfrich  refractometer  is  given  by  the  relation 

n  =  \/N2  —  sin20, 

where  N  is  a  constant  of  the  instrument  and  0  is  the  measured 
angle. 

If  N  =  1.62100  ±0.00005  and  0  =  45°,  approximately,  howpre- 
cisely  should  0  be  measured  in  order  that  n  may  be  reliable  to 
0.1  per  cent.? 

69.  Given  two  resistances  A  and  B,  each  known  to  -fa  per 
cent;    A  =  2  ohms  approximately,  B  =  l  ohm  approximately. 


PROBLEMS  97 


If  settings  on  a  slide  wire  bridge  are  IA  =  660  mm.,  and  Zs 
mm.  approximately  and  these  are  each  known  to  ±0.2  mm., 
compute  the  deviation  measure  of  n. 

1AB—1BA 


n  — 


A  —  B 


70.  What  considerations  determine  the  precision  with  which 
the  constants  in  the  equation  of  a  straight  line  may  be  ob- 
tained  by  the  graphical  method?    With  plotting-paper   10 
inches  on  a  side  and  ruled  in  twentieths  of  an  inch,  explain 
what  the  extreme  limits  of  precision  attainable  are. 

71.  What  is  a  residual  plot,  and  what  is  its  use?     Explain 
fully. 

72.  How  would  you  determine  by  a  graphical  method  the 
value  of  the  constants  a,  b,  and  c  in  the  equation 

y  =  a  +  bx  +  ex2 
from  a  series  of  values,  x\t  y\,  xz,  y2,  etc.? 

73.  What  is  a  logarithmic  plot  and  to  what*  class  of  prob- 
lems is  it  applicable?    Explain  fully  its  use  by  an  illustration, 
first,  using  rectangular  co-ordinate  paper,  and  second,  using 
logarithmic  paper. 

74.  The  heat  H  generated  in  a  coil  of  wire  in  a  given  time 
varies  as  the  square  of  the  current  /.     How  would  you  test 
this  relation  graphically  by  a  series  of  determinations  of  H 
and  If 

75.  How  would  you  test  graphically  the  law  that  the  de- 
flection of  a  beam  loaded  at  its  centre  and  supported  at  its 
ends  is  proportional  to  the  cube  of  its  length? 

76.  From  a  series  of  measurements  on  the  time  of  vibration 
and  corresponding  length  of  a  pendulum  explain  how  you 
could  obtain  the  mean  value  of  g  by  treating  the  data  graphi- 
cally,  i-i 

77.  From  a  series  of  measurements  on  the  strength  of  cur- 
rent 7  flowing  in  a  circuit  of  resistance  R  to  which  a  variable 


98  PROBLEMS 

(measured)  electrometer  force  E  is  applied,  explain  how  you 
would  find  the  mean  value  of  R  by  treating  the  data  by  a 
graphical  method. 

78.  Suppose  a  current  /  is  measured  by  a  tangent  galva- 
nometer for  which  I=K  tan  0>*   From  a  series  of  values  of 
7  and  corresponding  values  of  0  explain  how  you  would  find 
K  by  a  graphical  method. 

79.  The  formula  for  the  focal  length  of  a  lens  is 

iJUi. 
/   P    P' 

From  a  series  of  measurements  on  p  and  pf  explain  how 
you  would  find  the  mean  value  of  /  by  means  of  a  graphical 
method. 


APPENDIX. 


TABLE  I. 
MATHEMATICAL  CONSTANTS. 


Quantity. 

Value. 

Remark. 

Common 
Logarithm. 

c 

2.71828 

Base  of  Napierian,  natural 
or  hyperbolic  logarithms. 

0.434295 

loglO  € 

0.434295 

Factor  to  multiply  Napier- 
ian logs  to  convert  into 
Common  logs. 

9.637784 

1 

loglO  € 

2.30259 

Factor  to  multiply  Com- 
mon logs  to  convert  into 
Napierian  logs. 

0.362216 

7T 

3.14159 

Ratio  of  circumference  to 
diameter. 

0.497150 

1/7T 

0.318310 

Reciprocal  cf  ir. 

9.502850 

X2 

9.86960 

Square  of  IT. 

0.994300 

V^ 

1.77245 

Square  root  of  x. 

0.248575 

1  radian 

57°17'45" 

57°.2958  =  206265"  =  arc 
equal  to  radius. 

1  degree 

^  =  0.017,4533  radian 

100 


APPENDIX 


TABLE  II. 
APPROXIMATION  FORMULA. 

It  frequently  happens  in  a  computation  that  a  factor  of  the 
general  form  (1  ±  a)n  enters  wjaere  n  is  a  constant  and  a  is 
a  quantity  whose  numerical  value  is  small  compared  with 
unity.  In  such  cases  the  approximate  value  of  the  factor 
given  by  the  first  two  terms  of  its  expansion  -may  usually  be 
substituted  in  place  of  the  factor  itself  without  introducing  an 
appreciable  error  in  the  result,  and  the  computation  becomes 
thereby  decidedly  simplified.  If  the  factor  is  of  the  form 
(m±  a')n  where  a'  is  small  compared  with  m,  it  may  be  written 


-  1  =mn  (1  ±  a)n  and  so  reduced  to  the  first  form. 


Table  II.  contains  the  approximate  forms  of  the  factor  for 
several  values  of  n,  together  with  the  error  which  would  be 
introduced  by  using  the  approximation. 


Factor. 

Approximate 
Form. 

Resulting 
Error. 

Computed  Error 
if  d  =  0.01. 

(1  ±  a)* 

l±a 

n(n-l) 

2 

(1  ±  a)2 

1  ±2a 

a2 

0.0001 

(1  ±  a)3 

1  ±  3a 

3a2 

0.0003 

(1  ±  0)4 

1  ±4a 

6a2 

0.0006 

(1  ±  a)J 
(1  ±  a)i 
(1  ±  a)-* 

1  ±  \a 
1  ±ia 
1  Ta 

—  |a2 
-i<*2 
a2 

—0.00001 
—0.00001 
0.0001 

(1  ±  a)-» 

1  =p2a 

3a2 

0.0003 

(1  ±  a)-» 

Iqp^a 

fa2 

0.00004 

(1  ±  a)-* 

1  Tia 

fa' 

0.00002 

For  (1  ±  a)  (1  ±  6)  (1  ±  c)  use  (1  ±  a  ±  b  ±  c). 


For 


I  mi  ma  use  mi  0 — -  when  mi  and  m%  are  nearly  equal. 

m 


APPENDIX 


101 


TABLE  III. 
SQUARES,  CUBES,  RECIPROCALS. 


No. 

Square. 

Cube. 

Recip. 

No. 

Square. 

Cube. 

Recip. 

1.0 

1.00 

1.00 

1.00 

5.5 

30.3 

166. 

.182 

1.1 

1.21 

1.33 

0.909 

5.6 

31.4 

176. 

.179 

1.2 

1.44 

1.73 

.833 

5.7 

32.5 

185. 

.175 

1.3 

1.69 

2.20 

.769 

5.8 

33.6 

195. 

.172 

1.4 

1.96 

2.74 

.714 

5.9 

34.8 

205. 

.169 

1.5 

2.25 

3.38 

.667 

6.0 

36.0 

216. 

.167 

1.6 

2.56 

4.10 

.625 

6.1 

37.2 

227. 

.164 

1.7 

2.89 

4.91 

.588 

6.2 

38.4 

238. 

.161 

1.8 

3.24 

5.83 

.556 

6.3 

39.7 

250. 

.159 

1.9 

3.61 

6.86 

.526 

6:4 

41.0 

262. 

.156 

2.0 

4.00 

8.00 

.500 

6.5 

42.3 

275. 

.154 

2.1 

4.41 

9.26 

.476 

6.6 

43.6 

287. 

.152 

2.2 

4.84 

10.6 

.455 

6.7 

44.9 

301. 

.149 

2.3 

5.29 

12.2 

.435 

6.8 

46.2 

314. 

.147 

2.4 

5.76 

13.8 

.417 

6.9 

47.6 

329. 

.145 

2.5 

6.25 

15.6 

.400 

7.0 

49.0 

343. 

.143 

2.6 

6.76 

17.6 

.385 

7.1 

50.4 

358. 

.141 

2.7 

7.29 

19.7 

.370 

7.2 

51.8 

373. 

.139 

2.8 

7.84 

22.0 

.357 

7.3 

53.3 

389. 

.137 

2.9 

8.41 

24.4 

.345 

7.4 

54.8 

405. 

.135 

3.0 

9.00 

27.0 

.333 

7.5 

56.3 

422. 

.133 

3.1 

9.61 

29.8 

.323 

7.6 

57.8 

439. 

.132 

3.2 

10.2 

32.8 

.313 

7.7 

59.3 

457. 

.130 

3.3 

10.9 

35.9 

.303 

7.8 

60.8 

475. 

.128 

3.4 

11.6 

39.3 

.294 

7.9 

62.4 

493. 

.127 

3.5 

12.3 

42.9 

.286 

8.0 

64.0 

512. 

.125 

3.6 

13.0 

46.7 

.278 

8.1 

65.6 

531. 

.123 

3.7 

13.7 

50.7 

.270 

8.2 

67.2 

551. 

.122 

3.8 

14.4 

54.9 

.263 

8.3 

68.9 

572. 

.120 

3.9 

15.2 

59.3 

.256 

8.4 

70.6 

593. 

.119 

4.0 

16.0 

64.0 

.250 

8.5 

72.3 

614. 

.118 

4.1 

16.8 

68.9 

.244 

8.6 

74.0 

636. 

.116 

4.2 

17.6 

74.1 

.238 

8.7 

75.7 

659. 

.115 

4.3 

18.5 

79.5 

.233 

8.8 

77.4 

681. 

.114 

4.4 

19.4 

85.2 

.227 

8.9 

79.2 

705. 

.112 

4.5 

20.3 

91.1 

.222 

9.0 

81.0 

729. 

.111 

4.6 

21.2 

97.3 

.217 

9.1 

82.8 

754. 

.110 

4.7 

22.1 

104. 

.213 

9.2 

84.6 

779. 

.109 

4.8 

23.0 

111. 

.208 

9.3 

86.5 

804. 

.108 

4.9 

24.0 

118. 

.204 

9.4 

88.4 

831. 

.106 

5.0 

25.0 

125. 

.200 

9.5 

90.3 

857. 

.105 

5.1 

26.0 

133. 

.196 

9.6 

92.2 

885. 

.104 

5.2 

27.0 

141. 

.192 

9.7 

94.1 

913. 

.103 

5.3 

28.1 

149. 

.189 

9.8 

96.0 

941. 

.102 

5.4 

29.2 

157. 

.185 

9.9 

98.0 

970. 

.101 

102 


APPENDIX 


TABLE  IV. 
FOUR  PLACE  LOGARITHMS. 


•3  2 

PROPORTIONAL  PARTS. 

i 

O 

I 

2 

3 

4 

5 

6 

7 

8 

9 

£3 

j» 

I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

4 

8 

12 

17 

21 

25 

29 

33 

37 

11 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

075.5 

4 

8 

11 

15 

19 

23 

2G 

30 

34 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

3 

7 

10 

14 

17 

21 

24 

28 

31 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

3 

6 

10 

13 

IG 

19 

23 

26 

29 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

3 

6 

9 

12 

15 

18 

21 

24 

27 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

3 

6 

8 

11 

14 

17 

20 

22 

25 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

3 

5 

8 

11 

13 

IG 

18 

21 

24 

17 

2304 

2330 

2355 

2380 

2405 

2430  2455 

2480 

2504 

2529 

2 

5 

7 

10 

12 

15 

17 

20 

22 

18 
19 

2553 

2788 

2577 
2810 

2601 
2833 

2625 
2856 

2648 

2878 

2672 
2900 

2695 
2923 

2718 
2945 

2742 
2967 

2765 
2989 

2 

2 

5 

4 

7 
7 

9 
9 

12 
11 

14 
13 

16 

IG 

19 

18 

21 
20 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

2 

4 

6 

8 

11 

13 

15 

17 

19 

21 

3222 

3243 

3263 

3284 

3304 

3324  3345 

3365 

3385 

3404 

2 

4 

6 

8 

10 

12 

14 

16 

18 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

2 

4 

6 

8 

10 

12 

14 

15 

17 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

2 

4 

6 

7 

9 

11 

13 

15 

17 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

2 

4 

5 

7 

9 

11 

12 

14 

16 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

2 

3 

5 

7 

9 

10 

12 

14 

15 

26 

4150 

4166 

4183 

4200 

4216 

4232  j  4249 

4265 

4281 

4298 

2 

3 

5 

7 

S 

10 

11 

13 

15 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

2 

3 

5 

6 

8 

9 

11 

13 

14 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

2 

3 

5 

6 

8 

9 

11 

12 

14 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

1 

3 

4 

6 

7 

9 

10 

12 

13 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

1 

3 

4 

6 

7 

9 

10 

11 

13 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

1 

3 

4 

6 

7 

S 

10 

11 

12 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

1 

3 

4 

5 

7 

8 

9 

11 

12 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

1 

3 

4 

c 

ij 

6 

8 

9 

10 

12 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

1 

3 

4 

5 

G 

8 

9 

10 

11 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5515 

5527 

5539 

5551 

1 

2 

4 

5 

G 

7 

9 

10 

11 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

2 

4 

5 

G 

7 

8 

10 

11 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

2 

3 

5 

6 

7 

8 

g 

10 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

2 

3 

5 

G 

7 

8 

g 

10 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

2 

3 

4 

5 

7 

8 

e 

10 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

2 

3 

4 

5 

6 

8 

9 

10 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

2 

3 

4 

5 

6 

7 

8 

9 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

2 

3 

4 

5 

6 

7 

8 

9 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

2 

3 

4 

5 

6 

7 

8 

9 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

2 

3 

4 

5 

6 

7 

8 

9 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

2 

3 

4 

5 

6 

7 

8 

9 

46 

6628  6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

2 

3 

4 

5 

G 

7 

7 

8 

47 

6721  6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

2 

3 

4 

5 

5 

6 

7 

8 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

1 

2 

3 

4 

4 

,5 

6 

7 

8 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

1 

2 

3 

4 

4 

5 

6 

7 

8 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

1 

2 

3 

3 

4 

5 

6 

7 

8 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126  7135:7143 

7152 

1 

2 

3 

3 

4 

5 

G 

7 

8 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210  7218  7226 

7235 

1 

2 

2 

3 

4 

5 

6 

7 

7 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292  17300'  7308 

7310 

1  2 

2j  3 

4 

5 

6 

6 

7 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

738073887396 

1  2 

2  3 

4 

5 

6 

6 

7 

APPENDIX 

TABLE  IV. 
FOUR  PLACE  LOGARITHMS. 


103 


Natural 
numbers. 

O 

I 

2 

3 

4 

'5 

6 

7 

8 

9 

PROPORTIONAL  PARTS. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

55 

7404 

74U 

J741S 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

1 

2 

2 

3 

4 

5 

5 

6 

7 

56 

7482 

749C 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

755 

1 

2 

2 

3 

4 

5 

5 

6 

7 

57 

7559 

756C 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

1 

2 

2 

3 

4 

5 

5 

6 

7 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

770 

1 

1 

2 

3 

4 

4 

5 

6 

7 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

1 

1 

2 

3 

4 

4 

5 

6 

7 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

1 

1 

2 

3 

4 

4 

5 

6 

6 

61 

7853|7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

1 

1 

2 

3 

4 

4 

5 

6 

6 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

1 

1 

2 

3 

3 

4 

5 

6 

6 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

1 

1 

2 

3 

3 

4 

5 

5 

6 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

1 

1 

2 

3 

3 

4 

5 

5 

6 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

1 

1 

2 

3 

3 

4 

5 

5 

6 

66 

8195 

8202 

8209:8215 

8222 

8228 

8235 

8241 

8248 

8254 

1 

1 

2 

3 

3 

4 

5 

5 

6 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

1 

1 

2 

3 

3 

4 

5 

5 

6 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

1 

1 

2 

3 

3 

4 

4 

5 

6 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

1 

1 

Q 

2 

3 

4 

4 

5 

6 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

1 

1 

2 

2 

3 

4 

4 

5 

6 

71 

8513 

8519 

8525:8531 

8537 

8543 

8549 

8555 

8561 

8567 

1 

1 

2 

2 

3 

4 

4 

5 

5 

72 

8573 

8579 

8585:8591 

8597 

8603 

8609  8615 

8621 

8627 

1 

1 

2 

2 

3 

4 

2 

5 

5 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

1 

1 

2 

2 

3 

4 

4 

5 

5 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

1 

1 

2 

2 

3 

4 

4 

5 

5 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

1 

1 

2 

2 

3 

3 

4 

5 

5 

76 

8808 

881418820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

1 

1 

2 

2 

3 

3 

4 

5 

5 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

1 

1 

2 

2 

3 

3 

4 

4 

5 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

1 

1 

2 

2 

3 

3 

4 

4 

5 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9026 

1 

1 

2 

2 

3 

3 

4 

4 

5 

80 

Q031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

1 

1 

2 

2 

3 

3 

4 

4 

5 

81 

)085 

9090 

909619101 

9106 

9112 

9117 

9122 

9128 

9133 

1 

1 

2 

2 

3 

3 

4 

4 

5 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

1 

1 

2 

2 

3 

3 

4 

4 

5 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

1 

1 

2 

2 

3 

3 

4 

4 

5 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

1 

1 

2 

2 

3 

3 

4 

4 

5 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

1 

1 

2 

2 

3 

3 

4 

4 

5 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

1 

1 

2 

2 

3 

3 

4 

4 

5 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

0 

1 

1 

2 

2 

3 

3 

4 

4 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

0 

1 

1 

2 

2 

3 

3 

4 

4 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

0 

1 

1 

2 

2 

3 

3 

4 

4 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

0 

1 

1 

2 

2 

3 

3 

4 

4 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

0 

1 

1 

2 

2 

3 

3 

4 

4 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

0 

1 

1 

2 

2 

3 

3 

4 

4 

93 

9685 

9689 

9694 

9699 

9703 

708 

9713 

9717 

9722 

9727 

0 

1 

1 

2 

2 

3 

3 

4 

4 

94 

9731 

9736 

9741 

9745 

9750 

754 

9759 

9763 

9768 

9773 

0 

1 

1 

2 

2 

3 

3 

4 

4 

95 

9777 

9782 

9786 

9791 

9795 

800 

9805 

9809 

9814 

9818 

0 

1 

1 

2 

2 

3 

3 

4 

4 

96 

9823 

9827 

9832 

9836 

9841 

845 

9850 

9854 

9859 

9863 

0 

1 

1 

2 

2 

3 

3 

4 

4 

97 

9868 

9872 

9877 

9881 

9886 

890 

9894 

9899 

9903 

9908 

0 

1 

1 

2 

2 

3 

3 

4 

4 

98 

9912 

9917 

9921 

9926 

9930 

934 

9939 

9943 

9948 

9952 

0 

[ 

1 

2 

9 

3 

3 

4 

4 

99 

9956 

9961 

9965 

9969 

9974 

978 

9983 

9987 

9991 

9996 

0 

' 

1 

2 

} 

3 

3 

3 

4 

104 


APPENDIX 


TABLE  V. 

SINES,  COSINES,  TANGENTS. 


Natural.       ** 

Logarithmic. 

Sin. 

Cos. 

Tan. 

Sin. 

Cos. 

Tan. 

0.0 

0.0000 

1.0000 

0.0000 

—  oo 

0.0000 

—  oo 

0.5 

0.0087 

1.0000 

0.0087 

7.9408 

0.0000 

7.9409 

1. 

0.0175 

0.9998 

0.0175 

8.2419 

9.9999 

8.2419 

1.5 

0.0262 

0.9997 

0.0262 

8.4179 

9.9999 

8.4181 

2. 

0.0349 

0.9994 

0.0349 

8.5428 

9.9997 

8.5431 

2.5 

0.0436 

0.9990 

0.0437 

8.6397 

9.9996 

8.6401 

3. 

0.0523 

0.9986 

0.0524 

8.7188 

9.9994 

8.7194 

4. 

0.0698 

0.9976 

0.0699 

8.8436 

9.9989 

8.8446 

5. 

0.0872 

0.9962 

0.0875 

8.9403 

9.9983 

8.9420 

10. 

0.1736 

0.9848 

0.1763 

9.2397 

9.9934 

9.2463 

15. 

0.2588 

0.9659 

0.2679 

9.4130 

9.9849 

9.4281 

20. 

0.3420 

0.9397 

0.3640 

9.5341 

9.9730 

9.5611 

25. 

0.4226 

0.9063 

0.4663 

9.6259 

9.9573 

9.6687 

30. 

0.5000 

0.8660 

0.5774 

9.6990 

9.9375 

9.7614 

35. 

0.5736 

0.8192 

0.7002 

9.7586 

9.9134 

9.8452 

40. 

0.6428 

0.7660 

0.8391 

9.8081 

9.8843 

9.9238 

45. 

0.7071 

0.7071 

1.0000 

9.8495 

9.8495 

0.0000 

50. 

0.7660 

0.6428 

1.1918 

9.8843 

9.8081 

0.0762 

55. 

0.8192 

0.5736 

1.4281 

9.9134 

9.7586 

0.1548 

60. 

0.8660 

0.5000 

1.7321 

9.9375 

9.6990 

0.2386 

65. 

0.9063 

0.4226 

2.1445 

9.9573 

9.6259 

0.3313 

70. 

0.9397 

0.3420 

2.7475 

9.9730 

9.5341 

0  .  4389 

75. 

0.9659 

0.2588 

3.7321 

9.9849 

9.4130 

0.5719 

80. 

0.9848 

0.1736 

5.6713 

9.9934 

9.2397 

0.7537 

85. 

0.9962 

0.0872 

11.43 

9.9983 

8.9403 

1.0580 

90. 

1.0000 

0.0000 

00 

0.0000 

00 

00 

UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


E.NGINEEF 


ING   LIBHAMY 


LD  21-100m-7,'52(A2528sl6)476 


^ 


2662 


i- 


